If you have a 64bit laptop then PaleMoon is a good one (it's the 64bit version of Firefox that I use); I used to use Waterfox which was good but it has become quite buggy and the individual responsible for it seems to take ages to get around to fixing it, so probably best avoided at present.
In terms of terminology, Mozilla Firefox is an internet browser (as are Internet Explorer and Google Chrome), whereas a search engine is used from within an internet browser to find webpages that contain the information that you are seeking. You can have as many different internet browsers installed on your computer as you want, they are separate programs which don't interact directly with each other, just as you can run, for example, a word processor and be watching a video at the same time.