First off, I'd advise reading this thread here. Making the text font/size/colour different (even if it is bigger) can be actually harder to read, and that thread explains why. Also note that line spacing should occur between each time someone else begins speaking - i.e.
"Roooosseee!" said Drew's Roselia,
"Draaagonnaaaiirr!" said Dragonair, in reply.
should be
"Roooosseee!" said Drew's Roselia,
"Draaagonnaaaiirr!" said Dragonair, in reply.
This didn't occur all the time, so just take note of that. Also note that bolded comma should actually be a full stop, there.
On the beginning itself, it is a bit too short, I feel - and by that I mean that there's not much going with description or the short. Adding in more would certainly expand on what is here and make it more interesting, so consider doing that. For instance, with the above, you mention that they say their names, but not how they say them - enthusiastically? Angrily? Louder than the audience watching the battle? Furthermore, how did they act when they did that - did Dragonair wave its tail in the air? What about the Roselia? You can also describe them as well by doing so, and suddenly there's more to read, more happening, and it's easier to imagine as well.
Taking another instance:
"Right, give her an attract!" she ordered. Dragonair acted really cute, attracting the female Roselia to him.
"LEAF STORM!" Drew commanded.
The Roselia fell in love with Dragonair, but listened to its trainers command.
'actd really cute' is certainly better than 'it used attract', but some more would be better - how did it act 'really cute'? Describe the actions so we can visualise how this Attract attack was actually performed. How did Roselia fall in love as well - did she gape in wonder at the Dragonair, or consider giving him a flower? Consider how things occur with attacks and what the look like and so forth, and include those details in, and it'll be more interesting to read.
Also it is generally better to stick to using gender or 'its' rather than mixing the two - for instance, you refer to Dragonair as him and Roselia as female, but you also use 'its' which implies that its gender is unknown. Another issue of consistency is that you capitalised every other attack but 'Attract' itself was not capitalised - go one way or the other rather than a mixture of both, I suggest.
Watch out for small mistakes like this as well:
"Dragonair, Take the stage!" shouted May, smiling.
that 'take' doesn't need the capital as it is in the middle of the sentence.
If you apply that then this could be certainly be a neat story, but that's up to you. You can also check out other fics to see how other people write things for instance - at any rate, good luck with your story.