I have been looking throught different Gaige builds. Can someone tell me what are the best builds for her? Perks, gear, etc
The “best” is subjective: you want the one that works for you the best, which may not be the same as someone else.
I love elemental Gaige (middle tree, elemental damage buff COM/relic), but I actually have two versions: one anarchy, and one non-anarchy. Shock is easy to get gear for, since a number of blue unique side quest rewards in shock flavour will carry you a long way (Kitten, Rubi, Hail, etc.)
For the rest, I suggest you move over to the Gaige sub-forum. The build descriptions there will include descriptions of what the primary focus of the build is, whether it’s good for bosses, what gear is needed/works well, and why it’s put together the way it is. That way, you can find something that fits your preferred play style and take it for a test spin. There’s also an advice thread specifically for Gaige builds, so you can easily get help if you run into problems.