We call it "tonemes" in Norwegian, which, like lelisel said, isn't truly tones, but is still rather essential to learn in order to master oral Norwegian. This is never indicated in written language; thus, you have to learn the tonemes of the Norwegian words with 2 or more syllables where the stress isn't put on the last syllable. There are 2 tonemes in Norwegian:
Toneme 1: Present in most foreign words and loanwords, however, this also is highly frequent in common Norwegian aswell. _/ (starts low, then pitches up)
An example word: Bønder "peasants" (almost rhymes with "gunner" in English)
Toneme 2: Present in some loanwords aswell, but is mostly in the "most Norwegian" words. \/ (starts high, pitches slightly down and then goes up again).
An example word: Bønner "prayers, beans" (no English similar rhyme)
These example words may indeed be tricky to tell apart, and there surely exist more in Norwegian. I am very glad that I don't have to learn this as an adult, as it would've been way too hard to remember correctly all the time!
When it comes to whether you should choose Bokmål or Nynorsk, I would prefer Bokmål (lit. Book goal/tongue/language, with toneme 2). Approximately 85% of the population uses it, and Nynorsk (lit. New Norwegian, with toneme 2) is mostly kept alive as an official language through heavy promotion, following a lengthy debate over the course of the Norwegian language which lasted through the entire 20th century. Some would consider Nynorsk a more lyrical language, which I indeed think it is, but it has its flaws, and unless you would happen to settle in rural western Norway, you won't have much use for it compared to Bokmål. As a fluent Bokmål speaker you will have no problem at all understanding Nynorsk and vice versa.
Danish is way closer to Bokmål than to Nynorsk, and Danish is more or less mutually unintelligible with Nynorsk, whereas written Bokmål is very close to Danish. This has to do with the history of the Norwegian written language, as the modern Bokmål is based on the administrative (=Danish) language used in Norway, which was then made more Norwegian over the years to better suit the public language.
Norwegian words may be indefinitely added to each other to make endless chains; therefore there is no such thing as a longest word in Norwegian. My guess would be "menneskerettighetsorganisasjonene" (33 letters, the human rights organisations), which is the longest one in fairly common use. I have yet to see anyone use the word "minoritetsladningsbærerdiffusjonskoeffisientmålingsapparatur" used anywhere.
I did not mean in any way to denounce my fellow Norwegian lelisel, but I wanted to clarify and help you people a bit learning a great language!