Just started getting into Dominion and I'm wondering which spells should I consider choosing. Are there any go-to spells, like flash/ignite combo on SR? I've been using Revive/Exhaust for most of my roles.

Just always take revive (masteried as well) and pick another of your choice. They all have their niche. Except Clarity. Never take that.

Exhaust is probably #2 most powerful however.

Unless you're playing at high elo, I don't reccomend Revive. Ghost+Exhaust is a great goto combo. Having Revive, and utilizing it properly are two different things. Feeding an extra kill is no good, and a waste of a spell slot.

Best spells imo are Revive, Exhaust, and garrison.

I'm obviously not a high elo player, but I have been getting matched up in games where people are finally more concerned about capture points rather than kills. How am I supposed to use revive properly then? I usually use it right away if I die early on when trying to capture the top (haven't learned the names of these points...windmill, maybe?) point if I judge that I can make it back there in time to make a difference. Other times are when I'm defending bot and I use it to get back in the fight if they enemy who killed me is gonna end up capturing it if i don't use my revive.

Also, how would I use garrison? I haven't had the slightest idea of how to use this properly. Is this mostly a spell used for bottom? And when should I be using it? When too many minions are at my point and I can't clear them quick enough while the enemy is poking me to death? Or when there are a bunch of enemy champs at a point trying to capture?

Normally there are two garrisons on the team. One taken by the tank up top, and one by the bottom. The whole point behind garrison is it regens your point, and stops all capture upon use. If your enemy pushes a big creep wave to your tower you can garrison it to lessen the extent that the creeps take over.

As for Revive, there will always be killing. In order to cap top you HAVE to kill the other team, unless you are just going to sit there and watch them cap it.

tier 1: revive
tier 2: exhaust garrison ignite
tier 3: cleanse flash ghost

I would say
Tier S: Revive.
Tier 1: Exhaust, Garrison.
Tier 2: Ignite, Flash.
Tier 3: Ghost, Cleanse, Heal, Promote.
Tier F: Clarvoyance, Surge, Smite, Clarity.

Revive may not be the best at lower levels of play. I tend to use always revive now because everyone else on my team is always using it, but it could be an awful idea to be the only one on your team running revive.
The reason for this is that sometimes using revive really screws over your team because of the way respawn timers are set. Have you had this game where you're winning, and all of a sudden your team goes out of sync, and keep getting killed when they don't all go together? It is really more important to respawn with your team.
It's true that Revive is the one spell that may save a game by saving a point through using it. But maybe the game would have never reached that point if you had used Ghost 3 times earlier and ended up with 1 escape and 1 extra kill.