An awful lot, in theory. Alchemy is necessary for smithing extremely good weapons and helpful for fortifying enchanting. It's the easiest way to make money in the game. I avoid potions in battle but I still consider alchemy one of my core skills.
I'm not really sure, since I don't actually use them much, but I think they're probably overpowered. The time value seems to max out at 300 seconds (five minutes of realtime). I don't think there's a limit on potion use, so health potions effectively make you invincible. Using alchemy-enchanted gear you should be getting extremely strong potions - if they aren't worth using they're at least worth a lot of money so you can trade them for ultimate health potions at vendors.
Experimenter is 3 perks and not necessary, if you plan to go legendary you may want to take it but otherwise avoid it. Purity makes your potions have a lower value, and you can usually find the effects you want without needing it, so I think it's worth avoiding. Get all the rest if you want to focus on alchemy.
If you have Hearthfire, mora tapinella, scaly pholiota, and creep cluster - combine the three for a highly valuable fortify carry weight potion. In particular, plant 3 creep/3 scaly/5 mora in each garden, then plant 5 creep/5 scaly/8 mora in your greenhouse. I advise getting all the houses and building a greenhouse as soon as possible, that will give you a yield of 69 each run. If you sleep in each manor you can run a continual loop and always have fresh ingredients. If your load times aren't too high you can get 1,000 of each ingredient in about an hour.
If you don't have Hearthfire, blue mountain flower and blue butterfly wing, snowberry and purple mountain flower. Hanging moss and snowberry are plentiful and have some good combinations with many other ingredients. Juniper is easy to get a lot of (the Reach is littered with it), and it grows with hanging moss, but it doesn't make many high-value potions. Red mountain flower is again easy to get but low in value. Butterflies respawn in Riften when you enter and exit an area, so you can get them pretty easily (the blue ones are worth your time).
Most of the other plant ingredients are more scattered and harder to harvest in large quantities, so harvest when you see them and use them when they're convenient. Mushrooms are too rare to be a main crop, although some falmer areas have lots of glowing mushrooms (and chaurus eggs!). Animal materials and other exotic ingredients are basically impossible to get many of, just use what you find (or buy in stores).
Don't use daedra hearts for potions, save them for quests and smithing daedric gear.