First off, always trust your instincts! Fortunately, my family is nearby and I have some good friends at work, so I'm never short for a babysitter (not that hubby and I get to have "datenight" very often)!
My suggestions to find a sitter are: attend some kind of mom & kid play group (e.g. Gymboree) and ask around (the instructors, the parents). You could also contact some nearby daycares and ask if any of their employees do babysitting "on the side." Ask around at work--you may have a coworker looking for the same thing (a babysitter for date night). Ask around at any other place you attend regularly (church, school, gym, etc). Once you narrow it down to a few possibilities, find some way to open up the conversation--over a cup of coffee, or a kids' playdate. I'd want to get to know the person really well before leaving them with my kid, so you could invite them to lunch, meet up at the park, or have them over to your house for dinner. That's a great way for them to meet your family, see how they interact with your kid, and let them see how you interact with your kid.
Good luck!