It would be very difficult to get a guitar and amp for under $350. When I was looking to buy an electric I was going to get a Epiphone Les Paul Style one, but my friend who is VERY experienced in guitars and the like told me to either save up more $$, or buy used. I ended up spending more money than I wanted, but got an amazing guitar that plays and sounds like one 2-3x its cost.
As for an amp, look to buy one used as well. The newer Marshall stuff isn?t as good as the older stuff, especially in the lower spectrum. Look in the classifieds, as many people are looking to sell of old, unused guitar equipment at low prices. Make sure that you play everything before you buy it.
If you are looking to buy something on the internet, go a local guitar store and see if they have it in stock. Try it out there before buying it online. I really can?t stress that any more.