I know it's not very romantic, but talking about financial issues with your partner will save a lot of heartache down the road -- and this is even more true if you and your partner are not married.
First and foremost, it is probably not a good idea for unmarried couples to merge all bank accounts. However, a joint checking account to pay bills or buy groceries can be a good start to test how you both handle dealing together with money. In a more committed relationship, you can try opening a joint savings account for a specific goal, such as an upcoming vacation or your wedding.
Here are some tips for unwed couples who live together:
1. Schedule a time to talk about money.
2. Disclose how much each of you make and all debt that you carry.
3. Determine who will be responsible for each bill.
4. Plan for the worst. Who will stay in the residence after a breakup? What about division of personal property? What if you have a pet together? (If you have children together, obviously you're in a whole other ballpark and most of this does not apply.) You can make these decisions formally (with a contract, for instance) or informally.
5. Keep finances separate -- with the exceptions discussed above.