No. It is not a crime to investigate someone's IP address, or try and locate them. So you won't be arrested for that. Some of the methods that could be used to trace an IP, however, may be illegal.
There are a few ways you can get someone's IP address- usually the best way is, if they send you an email, you can usually get their IP out of the message header (most email providers have a way to show the header). The following site will analyse the header:
http://whatismyipaddress.com/trace-email
Once you have the IP, you can have it analysed in publicly available websites. Here's one:
http://whatismyipaddress.com/ip-lookup
These websites will reveal a lot of publicly-available information about an IP address, such as what ISP company it is registered to. They will also tell you where the ISP is located. That will almost always be in the same country as the IP itself is, although beyond that, the location data is often not very precise. It's usually (but definitely not always) in the same city as the ISP, however it's rarely any more accurate than that. It won't reveal their location, name, or phone number, or any of that.
Anyway, all of that is legal and above-board, and you can do that with no worries at all. For fun, you could do it on yourself, or an email from a friend, to see how accurate it is in your case.
Beyond that, though, there's not much more you can do legally. The only way to get specific information about an IP (such as home address), is to get the ISP to check their records and give that information to you. Pretty much no ISP will do this unless you have a court order to get access to that information, which you'll only be able to get as part of a court case or police investigation. So there is no way for most people to legally get someone's home address from their IP.
So that's more or less it. It's not a crime to track an IP, and the police won't arrest you for investigating an IP. However, there is no legal way to trace it down to a home address or anything like that. But it's not a crime in and of itself to investigate an IP address.
And obviously, if you tried something like, say, hacking the ISP, or hacking the target IP to get more information, that's obviously illegal and you could definitely get arrested for that.