With good equipment, a little experience and good maps, you can wander all over the Jotunheimen. Cairned trails take you up the main valleys, but with open country in every direction, you can trek at will. Sometimes the way you pick can get quite steep. You are in the Home of Giants, afterall.
There are many glaciers in the Park. Some extend for miles - Smorstabbreen, Veobreen, Memurubre. Crossing glaciers requires a knack for finding and avoiding crevasses - sprekken. Streams need to be forded, as well. Nothing quite as fun as walking 50-100 meters across a knee-deep glacial stream whose icy beginnings you can see just beyond;-\
There is a boat service on Bygdin and Gjende which allows people to get into the wilderness of Jotunheimen quickly. You can also send your pack ahead and walk the trails unencumbered. I have read where Norway is one of the most crime-free areas of the World. A study showed that of different countries, only in Norway were five 'lost' wallets returned to the owners. That said, I put my pack on the boat from Memurubu to Gjendesheim - with my Norwegian friends. By the time we had ambled over the Besseggen to Gjendesheim, my pack - with passport, a bit of money and priceless films taken from my treks - was stolen. The packs are just thrown out on a dock and you pick yours out. Someone else picked my out before me;-\ Be forewarned. It is not always easy to convince the Marines at the Embassy, you are who you say you are!
Driving along Sognefjellet - road 55, you can expect snow on the road even as late as early August. And close to its highest point, 1434 m a.s.l. new snowfalls can be expected even in mid-summer. We didn't have that problem in late August, but we could see snow lying quite near. If you are hiking, good boots would come in handy and so would walking sticks, to check the firmness of the ground or the snow patch before you step on it.
If you go hiking in the mountains of Jotunheimen, you must rely on yourself and your hiking experience only. The mountains are treacherous: the weather can change from hour to hour, the glaciers are not safe to walk on, if you stray from the marked routes, which is perfectly legal, you may have to cross streams, and, last but not least, you may easily get lost. You may think that in case of an emergency you can always use your mobile to call for help. Yet the mountains block the radio waves and, unless you have climbed one of the highest peaks, you might not get the signal at all. For environmental reasons all link stations are located outside Jotunheimen. It seems to be simpler from the main road that we travelled along - you can see my husband sending an SMS from our stopping place. I preferred to admire the scenery. If you look at the road sign to his right, you will realise how steep the roads can get - the 8% gradient is just the average.