Be in good shape. Know your equipment well. Do not bring more than necessary! Think about what you can do without. Mostly half of the cooking set can be left home. Do you need two pots? Do you really need a kettle? Etc.
You will need clothes for all kinds of weather but not one pair of trousers for sun, one for snow and one for in between. One pair of trousers and a pair of shorts or even just a pair of zip-offs will do. If it gets very cold – which it mostly does at night, you can build up with layers like, long johns, trousers, rain clothes etc.
Take care of the sun! It can be very, very tough up there. Remember you are outdoors most of the time. Bring a good sun blocker and leave all your deodorants and other urban luxuries at home.
Plan your trip carefully and be prepared to change plans if something unforeseen comes your way. If it is your first trip it is a good idea to stay on the marked common tracks like the famous Kungsleden. If you go away from the common tracks you must be prepared for crossing streams and rivers in often very cold water. This can be dangerous at times and many trekkers have lost their lives here.
Trekking in the Lapland fjells is not dangerous if you use your common sense. I have crossed a lot of rivers, been into quicksand, fallen off rocks into the water, climbed some steep fjell sides etc. if I have ever been in great danger or taken unnecessary risks I can't say. I have been forced to turn back and find another way because of too much water in a river, too much snow and crevasses or just bad weather. Maybe it has not always been necessary but I don't want to know. I always try to take care without being too mothered.
It is always a good idea to bring a good small first aid set.
There is one extra very important tip! Avoid using blue jeans – they are too heavy and will never dry once they get wet.
And one more advice – June, July and part of August can be very troublesome with mosquitoes, midges and flies, so bring good repellents. The Swedes have a lot of good ones like Nordic Summer and Jungle Oil and last year I tried the Smidge Repellent that I got from a good friend in the UK – works very well.