What are some recommended cordless trimmers/weed-eaters?
I own this cordless string trimmer: Black and Decker NST2018 ($100)

It comes with two batteries, so if the first one goes dead, you can pop in the second one and keep going.  (Always keep the spare one in the charger!)

I am happy with its performance.  My lot is 0.35 acres.  I use it to trim along my driveway, sidewalk, swing set, and a few flower beds.  I don't have a fence around my lot, or anything that requires extended use.  My usual trimming routine takes ~30 minutes, not counting cleanup.  I can usually finish the trimming in a single battery charge.  If I have neglected the yard for a while, it might take 1.5 battery charges to do everything.

It has an auto-string feed, which hasn't failed me yet.

You can rotate the trimmer 90 degrees to do edging work.  I do this along the edge of my sidewalk and driveway.  Makes a really "sharp" line, which looks nice.

It is pretty convenient to use.  No cords to drag around the yard, no engine to service.

However the battery does have its limits.  It starts out with decent power, but as the battery starts to wear down in the last few minutes, the motor RPMs will decrease.  It will still trim, just not as well.  If I had a larger lot, or a fence, or more things to trim around (garden, lots of landscaping features, etc), I would probably need something more powerful.  If your trimming routine is similar to mine, it should be OK for you.

My string trimmer is part of a larger Black and Decker 18V Yard System.  I also own the hedge trimmer and leaf blower from the set.  Since the batteries are interchangeable, I have plenty of spares and chargers.  Do you need other yard power tools?  If so, consider getting a set with interchangeable batteries.

When I got my yard tools, the Lithium Battery stuff was still pretty new, and no one was offering an interchangeable set.  Now there are at least a few on the market, so you may want to consider them over what I have.  Have never used one though, so I can't make a specific recommendation.