Practice. It will get better. I like to practice things in grassy fields, cuz when I go over it doesn't hurt as much and cuz the irregular surface adds to the effort while I practice.
(getting bottles in and out of cages riding in the grass makes doing it while riding on asphalt seem much easier, for example!)
"If Americans want to live the American Dream, they should go to Denmark." - Richard Wilkinson
Keep you weight loose and not leaning on your handlebars. Concentrate on looking quite a bit forward and focus on that straight line way out there, i.e. if you are riding down the road where there is a white line (and not too much traffic) ride the white line. You cannot look down real close to the bike to do that, you have to look quite a bit forward along the white line and all of the sudden you will be riding the white line Your bikes goes where you are looking. And, of course, practice just like everyone says.
spoke
It's a balance thing. At around 8 mph (for most bikes) it gets much easier to balance, and the bike is less twitchy. At walking speeds, most bikes will feel twitchy. If you practice pedaling smoothly at low speed, work on no hands riding and other balance exercises, it'll get a lot easier.
Most dogs walk at around human walking speed, but can run at least a bit faster than a human run speed. So it's ok to go a bit faster than a walking pace. Just keep an eye on your dog to make sure she's comfortable, since she needs to build up to distances just like you do.
Make sure your feet are going in constant circles and avoid "pedal steer".
Then again, I gave up and only walk my dog on my trike where I don't have to worry about it, so mine might not be the best advice![]()