This is an often-used bookmarked site for my early characters:
http://skyrim.melian.cc/?cmd=cmdSkyrimAlchemyWizard
You might scroll down the list of ingredients, just to become familiar with what you can actually use. Might find some surprises.

Two quick combos that I've committed to memory:
- BLUE mountain flower and wheat (fortify health & restore health)
For early levels it's probably no accident that blue mtn flower is found all along the road near Whiterun, and there's plenty of wheat to grab at the nearby farms right there.
- mudcrab chitin and charred skeever hide (cure disease / resist poison / restore stamina)
The fastest way to level up alchemy is to make expensive potions. Poisons, especially of paralysis, do well. Also multi-effect potions will bump the bar up faster.... don't worry at this point whether it's a good AND a bad effect potion as you can still sell it (just don't drink it in battle!).
You can (make a fortify barter and then) buy out all the ingredients at an alchemy shop... the shopkeeper will then have plenty of money to buy all your dodgy potions that you can't use (but that have really helped to bump up your alchemy).
But there are ingredients everywhere in the world, especially if you don't fast-travel..... different ingredients in different areas (snowberries in snowy areas, tundra cotton on the tundra, jazbay and creep cluster and dragon's tongue in the swamps, etc.).