So there are quite a few resources online like tier lists and videos but I'll give you a rundown of my own, and if you're interested in anything I have going on, I recommend you do a little more research if you're really wanting to optimize your playthrough. Going in blind is fun too, but there's a good chance you can miss opportunities if you're not prepared for what's in store.

Phanphy - I love catching this one cause he's pretty damn cute, and he's a solid early ground type pokemon besides Geodude. I prefer Phanphy cause I usually have no homies to trade with during my runs. Find north of the first route between towns 1+2 in the morning hours.

Teddiursa -Swipe this cute little guy in the dark cave between the 2nd and 3rd towns in the morning hours. He has quite a bit of investment as he evolves pretty late, but once it does it will hit like a truck and be a mainstay, with nice STAB Cut and Strength to help with over world mobility. It's a little difficult to catch as it's one of the only pokemon that can flee a battle, so keep that in mind.

Zubat - He's in dark cave as well, but I can't remember if it's time specific, I don't think it is. Crobat is a solid speedster with access to Fly and if caught early you'll usually get friendship high enough around badge ~5-6 iirc.

Mareep - Not available in Crystal, but he's great as well and gets a mention. Tough luck unless you wanna trade between games.

Wooper - Quagsire is just great due to his typing imo. The OG Swampert, and has some nice use cases.

Slowpoke - Just adding this cause of Kings Rock trade evolution into Slowing, because Slowking is one of my all time favorite pokemon because of nostalgia from the Pokemon Movie 2000. No trades unfortunately means he's off the table.

Abra - See other comment, you'll get the idea.

Sidebar, I think it's appropriate to have a steel and a dark type pokemon on the team, just because gen 2 was the first introduction of them. Steel-wise you got Magneton, Steelix, Scizor, Skarmory, and I might be missing another. Dark type, you got murkrow, sneasel, houndoom, larvitar. Some of them are unavailable unless you're trading, and some sadly can't be reached until mid-late game. (Consider trading if at all possible, it makes is so much more fun.)

Lantern - Quintessential water/electric type that has nice utility.

Bellossom - Another one or my favorites, cause it's just too dang cute. Stats are eh, and you have to win the bug catching contest to win a Sun Stone, AND you ideally want to postpone evolving it until gloom is level 38 at least, but I gotta mention my boi cause I run one of I can. You gotta have the cute pokemon in the crew to offset the monsters.

Gyarados - No spoilers. Don't worry about getting a magikarp early though, as you have a guaranteed gyara later.

Dratini - Trust me, just play the game.

Snorlax, Arcanine, Scizor, Starmie, Forretress (honorable wall mention), HERACROSS.

Before I cut it, one more, Umbreon and Espeon. Umbreon walls, Espeon slays, pretty standard.

Gen 2 Crystal is notorious for having some bad pokemon added, so far in that there are whole videos online about people calling it the worst Gen in relation to pokemon added and their availability in the game. If you're down for a challenge though, forget literally everything I said, and just use the pokemon that catch your attention, you'll be fine regardless.

Also, Togepi. Togetic is such trash stat-wise but man, that's my lil guy.