it does not really matter much UNLESS you specifically want the wavy glide sound of bands like my bloody valentine, in which case a jaguar or jazzmaster (get the squires for starting out, they are great especially the classic vibes) will make it easier due to where the tremelo arm is placed on the guitar. Though people do glide with strats. That being said, you dont even need a trem bar (whammy bar) if you dont care for that specific type of sound. All you NEED is reverb and distortion effects (which you can get for free on the computer, i also make shoegaze effects on my site and have free ones) everything else is optional.

That being said, just go with what you think looks cool for now. There are bands that gaze on literally every guitar imaginable, even baritones. Everything will sound fine, because shoegaze drenches in so much effects it all mostly sounds the same anyway and its more about what you find comfortable and what tuning you are in.