When we were first called as ward missionaries, my husband and I weren't given a lot of specific instruction. We were told that our main responsibilities would be friendshipping and teaching the Gospel Essentials class (not every ward missionary will teach this class, this is specific to our ward at this time. Often the class is taught by the ward mission leader or a seperately called teacher). The Church handbook of instructions says that we are supposed to find people to teach and assist the full-time missionaries in teaching, but doesn't give a lot of specifics, either. What EXACTLY were we supposed to do??? We decided that we should come up with our own ideas to make the most of our time.
One of the first ideas we came up with was simple - attend activities! What a fun "assignment" (if you can even call it that!). We haven't always been the greatest at attending every activity in the past, but since we became ward missionaries we've gone to everything we can - even activities that we would never have gone to in the past. For example, our ward had a Scout fundraiser dinner last month. We'd never been to anything like that before, but in the spirit of trying to "get out there", we bought tickets for ourselves and my parents. My parents had to back out at the last minute to go out of town, which left us with 2 extra tickets. It turned out to be a really good thing after all, because on a whim I called one of my newly assigned visiting teaching sisters and invited her and her non-member husband to join us. To my (pleasant!) surprise, they came and we had a really fun night getting to know them.
Participating in activities is a good way to get to know non-members and less-active members in a casual setting. Keep an eye out for anyone who looks uncomfortable or awkward and see what you can do to help. Even if there aren't any non-members or less-active members there, you can always use the opportunity to practice getting to know people (there are probably ward members that you don't know well because they are recent move-ins, serve in an auxillary, etc)