If I purchase a Roku2 for my TV (2012 Insignia HD) and install, will I immediately have access to local television stations? I have an internet router (Belkin) for my computer. Or is there other costs?

No it doesn't, I have a Roku2 and a Roku3 so I can definitively state that you need to buy an antenna...that's what WILL work.

Unless your stations have apps designed for roku then you will not get the stations except possibly on your computer if they broadcast throught the internet. Roku has a more specialized format then just local television.