Start with a helmet. Never put your butt on the seat without your helmet on your head.
Cycling shorts and gloves aren't 100% necessary, but they do make a big difference. A cycling jersey instead of a T-shirt also makes a big difference.
You will be on the road bike for much longer times than when you go running. Buy a big tube of sun screen and use it, especially on your neck, ears, and thighs.
Shop around for a true road bike. Go to the local shops and see what they have that fits into your budget. Test ride the bike. Go to other shops and test ride as many bikes as you can. One will just fit and feel better than the others. Buy that bike regardless of the name on it. The business is so competitive that any two bikes at the same price point are very likely to be equivalent values.
Road bikes get flats. Get a mini wedge bag that attaches under the seat. Get a set of tire levers and a spare tube to put in it, about $3 each when you hit the sales just right. Get a pump ($15 and up). Get water bottles. Your bike may or not come with water bottle cages. I always have two on my bike ($5 each).
To be most effective, you will need clipless pedals and shoes. However, those can wait for a while if cash is an issue.
HTH
