There are two principle non-surgical options for birth control: hormonal and barrier methods. Sounds like you've already decided to steer clear of hormones, at least for a time.
The most common barrier options include condoms (male or female) or diaphragms, in combination with a vaginal spermicide. (Vaginal spermicides, available as film, foam, and suppositories, have a slightly higher failure rate than other methods when used alone.) We haven't yet found a female condom that is easy to use and reliable--let us know if you have a recommendation! Don't count on condoms that have been rattling around in a drawer while you've been on the pill: Freshness does count so the condoms aren't brittle.
You'll need to talk to your health care provider about a diaphragm, since it needs to fit you well.
And, of course, there are surgical options for your partner (vasectomy) or for you (tubal interruption). Again, a discussion with your health care provider will help you weigh those options.
You're smart to be thinking ahead about this!