First time re-grouting my shower

Yes I'd remove all the old silicone. They make grout caulk that you can buy in tubes that match your grout color. You can use that around your tub and inside corners/spots where regular grout has a tendency to crack.

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Also, does anyone have experience with premixed grout? If so, notice any difference?

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Just more expensive. Mixing you're own grout is super easy, just don't put a bunch of water in your bucket or you'll have soup for grout. Put a little water in, mix, sponge some more water, mix, sponge more water in, and repeat until you have a good consistency. If you don't feel like doing that, pre mixed is fine.

You need to remove old grout to a uniform depth over the entire area to be regrouted. At a minimum remove at least 1/2 of a tile thickness. If you regrout over old, it may fall out, and any mildew will grow through.