Buying a "real" internet connection will cost you more than a decent hosting plan or a even your own VPS and you will have the problems of availability, electricity, finding somewhere to put a server, etc. etc. I really wouldn't recommend "self hosting" at home.
If all you want to do is run your website then just go for a better hosting plan. For under $10 per month you can find accounts with 150Gb of disk space, 25 MySQL databases, etc. It's also highly likely that your existing provider will be happy to sell you a better plan and you won't have to do anything to your existing site.
If you want to do more and want full access to the server then look at some of the VPS accounts that are available around $30 a month, or as low as $10 a month if you're willing to commit to a long contract.
Congratulations on getting traffic / visitors for your website and I hope it grows for you!