I'm actually interested... Are they independent or should I start with the first one? If I can play either which one is the best? I know very little about the franchise...
You should play the series in this order:
1. Gothic
2. Gothic 2 + NotR
3. Risen
4. Risen 2
5. Risen 3
6. Gothic 3
7. Arcania
Reasons:
All games are storywise either unrelated or only somewhat related (ie. it's not really important) with the exception of Gothic 1 & 2
It all began with Gothic and it still is the best (except game mechanics, which are the best in NotR). Gothic 2 is a direct continuation of the story. Risen is a very similar game to Gothic and as close as modern gaming will get to it. All three are great games.
Risen 2 and 3 are fine games as well, just not much like Gothic anymore, you should still enjoy them nonetheless (except shit fucking minigames in R3 FUCK)
Gothic 3 is the worst and the least like Gothic (except Arcania), the worst combat, hugeass and totally empty world with a dozen NPCs with one liners making whole town etc, you might enjoy the hiking though.
Arcania because if you disliked any of the above, after Arcania you will praise other games to heavens.
Gothics and Risens put heavy emphasis on dialogues and writing, and exploration.
The setting, story and hence the writing are quite gloomy and sinister, albeit not too much and there are some jokes here and there in Gothic, Gothic 2 and NotR and Risen.
Risen 2 & 3 had dialogues written by someone else (Pratchett) and they are more like a 'comedy', lots of jokes, imo funny, but whether you'll like them or not will depend on your taste. It's not similar in any way to previous games.
Gothic 3 has pretty bad writing. Travel dozens of kilometers for a quest, arrive at necromance's tower, talk to him and he says "hello. dark times ahead. go fetch me some bones from cementery please" or some such.
Arcania is literraly the worst written game fucking ever.
Gothic 1-2 and Risens have fairly small worls filled with wonders and interesting places every few metres. All hand placed and the result is gorgeous. Many astonishing places that you will remember for a long time (this applies especially for Gothic 1&2).
Gothic 3 has a hugeass generated world as bland as SKyrim and after exploring hundreds of kilometres, visiting towns and capitals, you will remember nothing because all is bland and shit.
Arcania I don't even know.