I would suggest just using Paypal, some people will disagree because they may have had a spat with PP in the past but I have been with them since 1999 and never had an issue. You can put the PP buttons right on your website with little effort. Hope that helps.
Agreed! PP is also very good about investigating disputes and having your back as long as you are in the right.
You can use their Virtual Terminal to process anything manually, and as Niko said, you can easily integrate it with your site two ways.
One is to have it accept Visa, Mastercard, Amex, and Discover. As long as your credit is good you should not have a problem getting that set up.
The other is to have the PayPal payment button he mentioned for people to use PayPal Express Checkout. They can use the funds from their own PP accounts to complete a transaction.
If your credit is not the greatest, use method two first for a few months, then go for method one once you have built a little trust and they can see a history with you.
Go with a decent E-commerce software that you can use to host your site & cart. I could be wrong, but I can't imagine anything free being sufficient. It would suck to later find that what you are using does not suit your needs and then having to go through all the work of setting up shop again.
Most of those E-commerce companies have plans that start around $25.00 a month and you can increase your bandwidth and other features as needed.
Good luck!