New campaign start

On saints row the third, if I start a new campaign, will I have all of my costumes, weapons, and vehicles? I want to make sure before I start another campaign

Progress, equipment, vehicles, etc. are saved on a per character basis. Each new campaign means a new character starting from the very beginning.

The exception to this is any DLC you may have purchased. Those are unlocked for all current and future characters.

There is a way to do this if you're on the PC version of SRTT. Go to the Saints Row Modding Forums (or should that be fora?) and download the save editor. Once you've completed your game, save it and then open the save in the editor. There is a button on the right labeled 'New Game +', which resets the missions and activities, but allows you to keep your char, funds, guns, cars, respect levels & skills, and Saints Book activities & challenges. Save the file, and restart using Continue in the game.

The game effectively restarts from the beginning (the Bank Heist in Stilwater) and you can play through the missions again. It skips the bit where you do your char selection (you already have one selected) but in all other matters, it's effectively like playing the game through again with all your skills.

Obviously, use at your own risk and always make a backup before overwriting your save, but it works amazingly well for me and adds to my game experience.

No, you won't. Starting a new campaign means starting from scratch.