I’m not a tattoo artist, but given how small that is, I wouldn’t let him start working on it again in less than a month. As to the colors, you can’t judge a half done, unhealed tattoo. I doubt it will ever be as vibrant as that pic.
2-3 weeks isn't entirely abnormal for return visits, but I'd expect that they'd be working on parts of it that haven't been touched yet, and that you'd have at least a 3rd visit some time later to at least touch up older work.
First, I wouldn't do anything until you've had it for several months. I've seen a lot of tattoos made MUCH worse because people were in a hurry to "fix" them. Second, why would you get a tattoo from someone without much experience? It's usually a recipe for dissatisfaction.
Having said that, I like it and think it looks good. Lots of people have tattoo regret. The vast majority get over it. Wear that tattoo proudly.