Don't be overly paranoid. The internet is not that dangerous a place.
I'll give you some advice, but bear in mind, I'm not a security expert, I'm just a learn-by-doing-it yourself guy like you.
You can run perfectly safe without a router, but with a sensibel software firewall. Many will probably argue that point, but this I know for sure. It does demand a little bit of attention to how your PC performs and that you act with care on the net. I'm not saying you should bypass that router, it's just to give you an idea of the security risk.
It's a good idea to run a software firewall besides your router. It can limit the outgoing data flow to the net. Your router does not do that. This might help contain viruses to give you an excample.
If you have your pc connected to the internet without any firewall protection, there is a very good chance you'll be attacked and fast. You should have your protection setup before you connect.
About the rest of the post, I think you'll find it thoroughly covered in the tutorials. There are some very knowledgble ppl on this forum.
If you want my advice then.
1/ Get Kaspersky AV. Updating should be swift and there is 30 days trial. This in the best AV on the market.
3/ For a software firewall ZA free or buy a full subsciption.