In reply to A Longleat Boulderer:
First of all a warning that all of my experience in Svalbard is getting around in boats rather than on foot, albeit for six years, but for what it's worth...
There's a fair bit of unglaciated terrain around Longyearbyen that you could explore, and if you decide to do this then you should check out Barentsburg, an active Russian mining town. If you're there early summer then it might be worth trekking E towards Storfjord and Negribreen -- sea ice lingers there so you might see seals and bears and there is an unglaciated route.
If you wanted to walk to Pyramiden (which I personally think is a must see, way more interesting and creepy than Barentsburg) then you'll need to cross several glaciers, at least one of which is very heavily crevassed. There are local boat tours that will take you across though, and you can stay in the hotel there and explore this area instead if you wanted to. From here you could walk to Skansbukta (would be a multi-day trip from Pyramiden) which has incredible waterfalls up until mid summer, and you might see puffins.
If you're particularly interested in wildlife then Alkhornet is worth seeing as it's often teeming with Arctic foxes and their kits, and reindeer are very common too. You'll for sure need a boat to get over there from Longyearbyen though. If you want to see walrus then local boats do trips to Poolepynten, but again there's no way to walk there. Any time you're walking along the coast keep an eye on the water, especially in shallow bays, as you might see beluga. Definitely bring binoculars for wildlife, and in my opinion they are vital bits of safety kit for spying out bears in the distance and avoiding them too -- I never go ashore without them.
Isfjord doesn't have a particularly big concentration of bears by the standards of Svalbard but they are around. Take seriously the advice you are given on running your camps safely and also conducting yourself generally, as obviously it would really suck to get yourself into a situation where you have to shoot one. Your best defence is to never stop looking around.
"Here and There Svalbard" is a good app with topo maps for the whole archipelago and integrates with your phone's GPS. The bible for travellers is Rolf Stange's book; it's full of useful advice and hopefully will give you a fair few ideas for walks to do. It also has a lot of info on history, flora and fauna:
https://www.spitsbergen-svalbard.com/?page_id=8434
If you do consider a boat based trip PM me and I'll be happy to suggest some good places to visit outside Isfjord.