To get a server hosted (as in gaming server, media server, high traffic website server... etc) you would have to look at virtual private servers (VPS), semi dedicated and dedicated server plans that various web hosts offer.
Bytehost offer a VPS plan for $9.99. The features aren't all that bad.
A dedicated server costs anywhere from around $80 (semi dedicated are a bit less).
Website hosting for the common man is cheaper than all of the above. It comes with more bandwidth and disk space (most hosts now offer unlimited space - with fair use policies) but you have to share your server with quite a few people - which causes latency.
Bytehost offer a shared hosting plan for $6 I think - sorry I keep reffering to them but they're my current web host and I'ev got their website bookmarked. They also offer a free shared hosting service - with quite a few limits.
Any paid hosting worth its salt should of fer error logs and most (almost definatly server hosting plans) offer logs of the people who have visited or connected to your server. If they don't it wouldn't be that hard to set up a database and a script that stores the IP address of all the people to view your website.
A few gaming server softwares also come with their own IP address logging stuff I believe.