Camera creation

Ive been scouring places for information on transformations in order to make a decent moving camera to explore the vast empty void. So far ive found a few different methods:

One with quadernerions
One that simply calls rotate three time ( one for each axis) then translates the coords
One that uss the lookat function

My question is which method would make the most sense? Ive tried 1 and 3, and have seen many people reccomend against 1.

Also in regards to lookat(), the parameters are simple enough but i have a question about its application.

The eye coordinates are very easy because that can be thought of as the physical location of the camera. If im trying to make a freeform camera that can look around in 3d space, how do i get the point it should be looking at? Im guessing i will have to figure out some circle math? If thats the case how do i get it to translate into a point to look at?

And finally for the up vector, i should be able to just take the perpendicular vector of the arbitrary point im looking at, correct?

Thanks for the help