Wikipedia comes to the rescue, listing (almost?) all software packages that can be used for this:
I can't personally offer you anything in particular. I've used Avidemux, Kdenlive, VirtualDub, Cinelerra, (and Adobe Premiere), and I've found them all decent for what I was trying to do at the time, but they are suitable for different tasks and they are used differently, so YMMV.
I've also used Windows Movie Maker, and I wasn't happy with it (it could only output to formats I didn't want IIRC), but that was probably the first version ever, so it no longer applies.