As Sixty Fiver said -- the key to staying straight is all in where you're looking. Pick out a point far down the road. I'll bet that you can ride a straight line right to it. You will go where your eyes are looking. You'll also see road hazards -- i.e., broken glass, potholes, sewer grates, etc. -- long before you get to them and you can then gently steer around them instead of swerving at the last second.

Look down the road. As you get more comfortable looking further ahead you'll also get more comfortable with riding faster, as you'll find that you'll have plenty of time to react to anything in the road. And, as you'll quickly find, the faster the bike is going, the easier it is to balance, which inspires more confidence, which inspires the desire to go faster, which... But it all starts with looking 100+ feet in front of you. You'll see the big stuff and you'll learn to trust the tires will roll right over the little stuff.