Hi J.,
My 27 month old daughter is the same way, she's so far our only child, but it started as early as about 20 months. It's super frustrating and it's just a phase. I'd caution you against using labels, like bully, because soon it becomes an expectation (how to talk so your kids will listen and listen so your kids will talk is a great book with a section on labeling). anyway, what i've learned is that my daughter is a bit impulsive (hitting and pushing when she feels a little "caged") and also a very sensitive child (like if someone else even says as kindly as possible, oh no sweety we don't hit, there's a total meltdown. . . different than if mom or dad say it). and that i just have to be about 1/2 a foot from her at all times when playing or around other kids and anticipate the behavior, which i do by reading her body posture because it's become pretty apparent now when she might impulsively tag someone with a toy or push them over. some of it is asserting her place in everything, some of it is liking to see the cause and effect, some of it is i believe, territorial, and some of it is about personal space. the most important thing is keeping your little one safe as well as other children safe, and for the most part i try to opt out of a lot of kid activities, we do some, but not a ton, like no playgroups, or times when i'd leave her with other adults and children without me there. it's hard, but it's just a phase and you just have to keep talking through it and setting expectations BEFORE you get somewhere with kids, though it may take months for this to actually work to your advantage, it helps your child know that you KNOW he can be different and eventually he will be old enough to get it. sorry i can't give you a better plan.