Know your limits. The winter environment is perfectly capable of killing the unwary tourist who gets lost in the fells. The rescue service works well – each year several tourists are rescued and only rarely any serious injury is sustained – but taking your chances is not recommended.
If you plan to travel alone or, for example in your own car, remember that distances are great and getting help for any unexpected situation may take time. Plan accordingly; take extra warm clothes in your car and tell the hotel staff where you are heading and when you expect to come back. One more thing worth mentioning is the hunting season: Natives are usually very keen of hunting, and the start of the season draws most hunters into the wilderness. Potential dangers can be countered by wearing a red cap or some other easily identified garment, and staying away from areas where hunting is allowed during the season.
Otherwise, there are few serious dangers to your well-being. Tap water and even water of lakes and creeks is potable in most places, bottled water contains more harmful compounds than tap water and foods are almost without exception safe to eat. Crime rates are low and people are helpful and nice in general but noisy foreigners on Friday night in a local pub/discotheque might be sitting ducks for harassment in extreme cases; violent attacks by drunken male villagers. This is mostly problem of skiing centers. Probability to get robbed or getting any other harm is still extremely low.
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The local language is Finnish, but as in everywhere in Finland, you'll survive very well with English. Swedish, though official, is rarely spoken due to Lapland's inland location. Therefore, English is basically the sole option for those who don't speak Finnish. Sámi languages are spoken sparsely in norhernmost areas Sodankylä, Inari, Utsjoki and Enontekiö.