When the foal was about 4 weeks old I started taking her around the yard, and after about a week of that she proved that she’d glue her nose to my heel and follow her mom. I took her out on trails, trails I knew well and that didn’t stray that far from home. I didn’t halter her, I didn’t halter her much at all when she was really young…and not in any situation where she could panic or pull back…I understand that it is VERY easy for them to injure their jaw/poll/neck at that age and deal with stiffness their whole lives.
People at the H/J barn would have gasped in horror, but any of the draft horse/old skool guys I look up to wondered why I waited so long! They will often have a foal following mom while she’s in harness. You just have to remember that the foal has almost no stamina. 15-20 minutes MAX of following you riding. Choring with a foal is different, because you stop the hitch a lot.
Once the foal started to approach the 3 month mark, she got too bold for us to leave the yard. That’s when I really started her halter training in earnest, and started taking her on her OWN trail walks, areas she was familiar with, without her mom. My strategy of “you lead a few steps you get a cookie” worked well, and she learned to behave on the halter quickly. If I had kept her on her mom, and she was leading well, I might have started taking her back out on a lead…but she was getting so big (and fat) that I had to wean her before her halter training progressed that far.
ETA: you need to pick the trail carefully too. These were quiet, sleepy trails, no traffic on the way…no surprises.