I live at a place where the WiFi signal overshoots my house, and thanks to that I have to use dialup. My brother has an Android phone I think and he is able to browse the internet and stuff using 3G, which lead me to think about trying to get 3G. Could I buy a 3G router and have it broadcast WiFi around my house? I think you have to pay for 3G but I'm not entirely sure how it works. Any help or information would be much appreciated.
Yes, if you have 3G coverage you can buy a 3G router and it will act as any normal WiFi router (for example with cable access). All the clients will have internet access over 3G.
My advise would be: If your friend's phone is running Android 2.2 or higher, it has a built-in WiFi hotspot function. Try using this Android hotspot function and see if you can get good enough browsing speeds, otherwise using 3G as a primary internet connection is a bad idea and frustrating experience.
You will definitely have to pay for 3G. Check with your mobile phone providers for a cheap plan. They mostly have data-only plans available which include a certain traffic limit of e.g. 10 GB per month.
Be advised: 3G data plans can become very very very expensive when you shoot over the monthly limit. So make sure you buy enough traffic and watch it carefully.