Buy and sell stuff. That seems to raise my speech skill. Lots and LOTS of stuff.
When a bandit says "hold it, you need to pay a toll to pass", instead of killing them, negotiate. Every time you pass that way you can increase your speech skill. And like tobs says, buy and sell stuff. Lot's of stuff. You need to look at the perk tree for speech, because somewhere up that tree there is a perk which gives you the option to invest with a merchant. Until you meet the requirements for that perk, you won't be able to invest with a merchant. Your chosen bride may require a longer "courtship" than most of the marriage options in Skyrim.