In order to investigate efficiently, you need to isolate stacks and ensure each is working as expected.
It also requires to understand how "accessing internet" works.
I don't like this wording as it is not accurate enough, even if I suppose you mean "using my web browser to access internet".
Keep in mind that accessing internet can also be FTP, SSH, telnet, news and technology to support these different access types may differ.
Back to your problem: assuming you want to access internet with your browser, meaning HTTP protocol, I think you need first to define proxy architecture taking in account the fact that you are connecting to an existing network which provides real access to internet.
- Is this network directly connecting the web (from HTTP standpoint)?
- is there any transparent proxy somewhere on the path?
- is there any explicit proxy?
Depending on above answers, then you may have to work later on DNS but this is definitely not the first step, IMHO.