Going to avoid a grammatical review, in part because you've already received one and in part because I'm intensely lazy. Just a few quick tips, however.
1. Your first paragraph ends without any ending punctuation mark.
2. Punctuating dialogue. Dialogue is a tricky little jerk to get a hang of, so I really don't blame you. Because I'd rather just jump into commenting on the content of the story, however, I'm just going to offer up a link to a guide that should help you:
http://fictionwriting.about.com/od/writingexercises/qt/punctuation.htm
Just remember that a dialogue tag (or tag line) doesn't have to be "he said" or "she said." It can also be anything like "I replied" or "he continued." In short, it's anything that tells the reader who said something and how it was being said.
Overall, yeah, you'll want to proofread carefully. The fewer errors you have in terms of grammar/mechanics/whatever, the more your readers can focus on your story instead of the little trip-ups scattered throughout your chapter. If you need help, there's really no shame in
finding a beta reader to look over your work after you proofread it yourself but before you post it to a forum.
That being said, let's check out the meat of the fic, shall we? A lot of this will be a bit on the stream-of-consciousness side, so if comments seem like they're coming out of the blue, sorry about that.
To start things off, it feels like you could draw out that wait a little bit more. It just seems like you have Tonya waiting for no apparent reason because the second the reader arrives at the scene to observe what's going on, the door opens. She even lampshades it a little by saying that the people inside the lab opened the door just as soon as she thought about it getting opened.
However, this would be a prime opportunity to let us get to know a bit about the character herself. I'm not talking about having her say, "I'm
Tsukino Usagi, fourteen years old the main character, and I'm this many years old with this kind of hair color, et cetera." I'm saying more along the lines of, "I've been waiting here in the cold for God knows how long, and here's a bunch of other sentences that show just how impatient I am." Or something along those lines. Just, you know,
anything that gives her a voice right off the bat, that makes her seem like she's just another person with pretty normal emotions.
In other words, I guess you can say that this is a wordier version of what Astinus was trying to point out. First person is a great perspective to use, but in doing so, you need to be prepared to delve into a character's mind and really bring out their thoughts and emotions. They're the ones telling the fic, so naturally, the story would be peppered with their observations and opinions, not just a straightforward account of what happened.
Speaking of first-person POV, the switching. On the one hand, I'm interested in seeing both characters give their version of how events went down and what to make of different characters. Moreover, I'd like to thank you profusely for not taking the newbie author route and putting anything like [LEON'S POV] smack in the middle of the fic. (I'm not sure if you were aware of this, but headers like that can get annoying
very quickly because of how much they're like the literary equivalent of a brick wall at the end of a highway with no speed limit.)
On the other hand, you'll really need to make sure first person POV works for one character before trying to do a trade-off of POVs. As in, you'd need to make sure you're really drawing out the thoughts and emotions of one character before even trying to tell a story through switching POVs. The reason why is because if the only difference between either POV is just in what happens (as in, there's no difference in characters' voices and thoughts on what's happening around them), then there's really very little that a reader can use to tell one point of view apart from another. It all ends up being generic, which would in turn make the reader wonder why you didn't just use third person POV, you know?
However, I do have to give you the thumbs up for choosing a Pokémon most people would overlook as both kids' starters. There's something to be said for giving characters something besides the fandom's standards. For one, it means the reader isn't expecting every battle to be just a brawn-versus-brawn fight where characters' main strategies involve abusing type differences. Of course, this means that you'll need to remember to follow through and make sure your battles involving these Pokémon turn out interesting. On top of that, it also means that your characters aren't basically Ash Ketchum expies, and they'll probably need to take some time at the beginning of the fic to figure out how to use their Pokémon. (Because, well, neither Yanma nor Swinub are notorious powerhouses.) So, yeah, right off the bat, it shows you've got some creative potential going on, which can be a plus in your favor if played right.
Going to agree with Bay, though. Upping the minimum age limit tends to be a step in the opposite direction because so many authors do it for either no apparent reason (besides making the characters older for one reason or another) or for reasons that aren't exactly well-chosen in consideration of the canon (i.e., "because it doesn't make sense for ten-year-olds to wander around a region unsupervised"). Because of this, readers might give you a sideways glance about your fic until you give a really,
really good (read: logical) reason for upping the minimum age. After all, in other regions, no one has much of a problem with kids wandering around the countryside alongside Pokémon. Additionally, this also means that trainers who travel from Kanto, Johto, what-have-you will probably run into problems competing, and newbie trainers from Shocan who travel to one of the canon regions will have their teeth kicked in by twelve-year-old veterans. It's just a bit too much of a rift, you know? This is especially awkward if the term "crossroads of all the regions" means more than just the fact that Pokémon from the canon regions live there.
(Side note about Unova: While the main characters themselves are older than any other player characters, there's no indication that the minimum age for a trainer's license is any higher than in any other region. In fact, if we also take a bit of anime canon, we can safely say that the minimum age is still ten thanks to Shooti.)
I'm not sure I understand why a professor is sending them out to collect gym badges and become champions, why they were selected to do it, and why they have to work together to get it done. While I know you're probably planning on answering all of this later, it just seems a little weird because there's not much in this kind of quest that would imply some kind of link to Pokémon research (unless you were playing with a bit of Pokémon Special canon by saying there was something special about the badges/position of champion). Remember that in canon, Pokémon professors tend to send characters out on missions to help further their research. That's why you get an empty Pokédex – because the regional professor expects you to fill it to help their research along. Red in Special was on the same quest, alongside many of the other 'dex holders until they got sidetracked by the plot. Alternatively, you've got HGSS-verse Elm, who sends the character out as a trainer to experiment with walking Pokémon. If there is no mission (like in the anime), then the professor basically serves as a distributor of Pokémon, and there's no real link between them and the badge quest.
The reason why I say this is because I'm finding it hard to imagine why a researcher would need to recruit two teenagers to go on a badge quest, especially two teenagers who weren't told what they should be observing and what kinds of things they should be reporting back to Professor Mahogany. Also, it's a little awkward because only one of them can complete the quest he just sent them on (because only one of them can become the champion).
But you did end on a rather hopeful note. Aside from the oddities and the uncertain logic of the mechanics of your world, your characters
do have potential. It's interesting to see two characters forcibly paired up without being told prior to being assigned their mission that they would be and without meeting each other first. And, of course, Tonya's "I hate you" leaves me thinking that there's going to be hilarious shenanigans involving at least one character who may or may not go out of her way to make the other's life as unpleasant as possible. You've got two potentially conflicting personalities forced to be together to go on a journey, so I have to say that I've got high hopes for the character interactions throughout this fic.
So, that just means you'll want to be careful with the delivery. This means proofreading, naturally, but it also means being careful to work out some pretty good reasons for why things are the way they are. More specifically, because their mission is why they're thrown together, you'll want to make absolutely sure that part makes sense. Also, you'll want to make sure that the characters know and understand what they're supposed to be doing from the get-go because your character interaction will potentially hinge on the differences between how Tonya and Leon approach what they're supposed to do.
On top of that, as I've said, you'll really want to work hard at bringing out the voice of both characters. Let us get into their minds, and have their opinions come out as they're telling the story. These are two different people. Think of how two people in real life might tell the same story. It might help to liken it to the way you talk about people. Do you have someone you
really hate? Imagine that you're telling a story about them to your best friend. Now, try to imagine that person telling someone else a story about you. Liken the way your characters speak to how people you know speak, and first-person point of view might come a little easier to you. Basically, it's all about trying to make your story seem like it's being told naturally, just in a regular conversation or something along those lines. That's all.
In short, yes. Your character interactions sound like they could have potential, but you'll really need to be careful with the execution.