Well I can tell you what my setup is and what works for me. Currently, I record any gameplay footage using a screen recording software called Bandicam. It's not as convenient as a dedicated capture card but it does the trick and has plenty of features that will suit your needs.
Once I have the footage, I use Adobe After Effects to cut it up and edit. Now for somebody just trying to cut a bunch of clips together, After Effects is a bit overkill. However, for anyone looking to do some more complex editing such as adding designs and animations to your videos, I cannot recommend it enough.