If he stops to eat while you're riding, pull up (fairly sharply, you want his attention) on one rein and cue for forward movement. You can spank him on the rump, going for forward, or you can try irritating him by thumping him with your legs until his head comes up and he's doing what you want. Grazing reins work for a "quick fix," but don't really address the actual habit.
It's better to tackle this on the ground to really fix the habit, since if you keep having to pull his head up you may end up causing discomfort in his mouth from the bit. If he tries to graze while you're leading him, pull up sharply on the lead and get his head up. When his head is up, pet him and praise. I like to pair the pull on the rope with a verbal cue, either "Head Up!" or "Acht!", so that, over time, the verbal cue will replace the pulling.
As far as keeping him walking, look up Clinton Anderson's Cruising exercise. That really works for making the horse responsible for moving at the speed you want until you say otherwise.
Hope that helps, and welcome to the forum!