I haven't played Yellow, but I have played Red/Blue, and even made a hack of it </shamelessplug>, and I can tell you this.
1. The first Gym leader is Brock, and all of his Pokemon are ground type. Considering your only starter option is Pikachu, you won't be able to hit Brock at all with the attacks you have, and it will take you a while before Pikachu will learn something that does even sparse damage. Your best bet on taking him on is catching a Caterpie and leveling it up until it learns Confusion at level 12 as Butterfree. It hits for normal damage against the terrible special defense of Brock's Pokemon. Don't worry, they don't know any rock-type attacks, so you should be fine. I've also heard you can catch Mankey and Nidoran before Brock in Yellow, and they learn fighting-type moves early, but don't quote me on that.
2. Monnotonne said this already, but it needs to be said again: the Psychic type in Gen 1 is god-tier. Due to a glitch, ghost-type moves do not work on Psychic types, which wouldn't matter anyways because the only move affected is Lick, and that wouldn't be the best move to defeat a Psychic anyways. Not only that, but special attack and special defense share the same stat. This means that special glass cannons like Alakazam actually are VERY resistant to special moves, and Chansey and Tauros are some of the better special attackers in this game.
3. If you're desperate for a good list of Gen 1 Pokemon, go to Smogon's Red/Blue database and look for Pokemon under the OU tag, such as Rhydon, Tauros, Snorlax, Starmie, and Exeggutor. Though I'm not sure how helpful that is because the toughest Pokemon you'll have to battle is a level 65, and you can go all the way to level 100. I'm sure you can solo the game with your favorite Pokemon if you're leveled enough and have the right moves.
4. You technically can catch them all in Yellow, but only through glitches. You won't find Weedle, Ekans, other Pikachu, Meowth, Koffing, Jynx, Electabuzz, Magmar, or Mew in Yellow using normal methods. Look up the Mew Glitch on Bulbapedia if you're interested in a complete Pokedex without trading, but be careful: doing it incorrectly might freeze the game or even corrupt your save data.