I have live in Norway (a few months shy of a year) and, before that, lived for almost a year in Denmark. I have visited Sweden, though at that time I only had mostly forgotten school Swedish at my disposal. I can currently have reasonable discussions with people using any of these languages, though I occasionally have to ask what a strange word means, and I have problems expressing exotic concepts. I do not sound native in any of the languages and they blend together, with Norwegian dominating at the moment.
Dabbling or specialization
It is a bit unclear what you mean by learning a language.
If you want a bit of tourist language, then learn that for each separately. They will reinforce each other and you might notice some differences, too.
If you want to be able to read stuff to rough understanding, then I would recommend going with two: Swedish and either Danish or Norwegian bokmål. Bokmål and Danish share a lot, with fairly systematic ways of figuring out which words correspond to each other. Swedish uses different letters for ä and ö (not æ and ø), which is not a big deal, but also has different vocabulary. I have heard that svenska and nynorsk resemble each other, but my knowledge of nynorsk is passive and I can't comment on that.
If you want to be understood, then I would focus on one language and dabble a bit with the others. Then adjust based on the little bit you have learned about the other languages; do not use garpegenetiv or "hatten min" -structure, for example, outside Norway, and try to use a correct word for plural you. Danish pronunciation is famously challenging, but they understand my moderately danified Finnish accent and a jumble of different Scandinavian expressions, so they'll probably also manage with whatever you throw at them. At worst they'll switch to English.
If you want to understand others, you will have to do the work and listen to the different languages (and accents for Norwegian). Danish is, again, famously challenging, but with Swedish and Norwegian you can possibly get away with learning one quite well and dabbling in the other.
My experience suggests that the poorer one is with a language, the more even minor distractions (such as noise, strange dialect, related language, hurry, being tired) matter. This is why I would recommend learning one of the languages well, to increase your tolerance to real life circumstances, but also dabble with the others a bit to get over the initial shock of what they sound like. But this is mostly speculative.
One exotic option would be to learn all of them as the same language. You would never sound native in any, but communication would probably work. I do not know whether anyone has tried this and how it worked, but it would be an interesting experiment.
I'll also mention that unless you are quite good at picking up foreign pronunciation, the Scandinavians might have easier time understanding you than each other, in the sense that the Dane might understand you better than they would a Swede speaking native Swedish. This is irrespective of which language you are using or trying to use.
You might meet a study which says that Norwegians have the easiest time understanding other Scandinavians and the others have the easiest time understanding them. It is not clear to which extent this is due to the qualities of the language, and how much comes from history, geography, the two written standards, the multitudes of strong dialects and their social position, etc. So I would not deduce from that particular study that learning Norwegian would be the best for communicating with all Scandinavians.
It might be true, but that study is not a very good basis for making the argument.
I have not experienced anyone being offended. The relations between the countries are good and close, so that is not an issue. There are, of course, jokes, and jokey stereotypes, as usual between neighbouring countries. My Norwegian has been commented as having some Swedish accent a couple of times, but not with hostility.