There aren’t any campsites really. We stayed at the one in Galtur (http://www.camping-zeinissee.galtuer.at/), which got, really, really intimate on some nights, given that you are essentially camping in the garden of a chalet. There are loads down the valley but they are on the other side of the Silvretta Pass, which is a tollroad, and a fair drive away (http://www.galtuer.com/de/silvretta-hochalpenstrasse).
From the campsite in Galtur, you can roll down the hill on a bike path to Ischgl, then catch the bus back up to Galtur (has a bike rack) with your lift pass – there is one more lift at the top which saves you a ride uphill back to the campsite if I remember correctly.
There are plenty of bike trails (http://www.ischgl.com/en/active/cycling-hiking/mountain-bike/tours). We only spent two days riding there. First was a self powered day, and rode the Helibronner Hutte trail which is a straight slog straight up a mountain. Not recommended. We got a lift pass on the second day and explored the top of the mountain. There were some pretty naff bike park type trails up top, loose high alpine gravel and no flow. We had lunch at a restaurant the cable car station, which had the poshest toilets I have ever seen up a mountain – all marble and mood lighting. Ischgl likes to bill itself as a playground to the stars. Then we rode back down on the Velill trail. Which was pretty hair-raising to be honest, there were a couple of points where I felt out of my depth. It would have been nice to strike a middle ground, and with all the trails that are mapped, I’m sure it can be found.
I kind of liked the place. There are some nice walks from the campsite, and if you walk up to the reservoir, there is a little display which shows how they pump water across the Alps for the hydroelectric scheme (http://www.see.at/en/see-pians-information/bergbahnen-see/ecology/power-plant)
Make sure you don’t leave the ‘all inclusive’ card at the campsite – it gets you discount on all sort of things, including the cable car.