There are quite a few packages that do this. Many are geared toward the casual user who just wants simple sequencing of clips. I know very little about those packages.
The more capable packages for the PC (in a reasonable price range) include Adobe Premiere Pro, Sonic Foundry Vegas Video, Avid Express DV, and Pinnacle Studio.
I know nothing of Pinnacle Studio, but I've had several bad experiences with Pinnacle products. They have a reputation for buying into market share rather than developing it. Pinnacle Studio may be good, though.
Avid Express DV has the advantage of being Avid. That's important to longtime professional editors but not for someone starting out. It's also expensive.
Premiere Pro is the lastest version of the runaway market leader in terms of market share. Premiere is a program that people love to hate, but it is not as hard to use as people say and enjoys the best third party support out there. Like Photoshop, Premiere is the application (along with FCP on the mac) that you are most likely to get help with since so many people know it.
Vegas Video is a newcomer from a company that is more of an audio specialist than a video house. It has good audio tools (probably not important) and has a user interface that's slightly different than others. It's users swear it's the easiest to learn, easiest to use, and most powerful out there. In fact, VV users are fanatical about their software. I like VV but find only a couple of things noticably better. Those are offset by a few irritations. VV's author, Sonic Foundry, was bought by Sony and some have questioned its future viability. Sony already owned a high-end editing package and clearly won't want to see that cannibalized by VV though steady improvements.
If I were starting out I'd give serious consideration to VV. It is supremely easy to do simple crossfades and its builtin color correction is very good. I'd also consider Premiere Pro. It is the PC standard and it integrates well with Adobe's great DVD authoring software.