Horses start bolting when there's something unpleasant or scary about what you're proposing, for whatever reason. Please realize I'm not calling you a bad fur-mom or proposing that your horse hates you - that's not it at all. I'm offering perspectives that you can use to figure out what's going on. =) Also, realize first that your filly is 3 years old. That's not mature yet. Despite being 'green broke,' your filly has lots of mental room to create 'options' for herself. As soon as she finds out that she can make a decision about something, she'll try it. If it's correct to you, it'll be easy for her. If it's not correct for you, you might go along with it - but it'll be harder work than doing the right thing. For every discussion you have with your horse about the right/wrong thing, she will gain experience. Part of training is preparing 'experiences' to introduce your filly to. When you do this, she will have decisions to make, and you can help her find and keep the correct (easy) decision. This is also a good way to start establishing a real relationship with your filly.
So, bolting. Things to consider:
Horse is sore? (You're riding a 3 year old - have you studied how a horse's back develops and when the spine is mature? If not, there are lots of online resources. Usually 4-5 years old is okay to start doing more serious riding.)
Have you been making many demands (or assuming your horse is just 'okay' with everything) without realizing that you've overwhelmed him with new lessons and work? Does he just want a break from all of that? Does your horse not see you as a good leader? Does your horse respect your leadership and think it's a positive experience? Horses tend not to like 'absolute tyranny' or 'boot camp' situations.
And lastly: is she just taking advantage of being free and likes being free? I generally find this a poor perspective - it's too easy to say that without asking 'what have I done to make working with me fun, positive, and possibly yummy?"
Does your horse believe that you are chasing her away when you go to catch her? Are you walking straight behind her without allowing her to come to you, or letting her get 'curious' about what happens if she comes up to you? Are you making approaching you enjoyable?
(These are all personal questions - no need to answer them here. Just ponder them and see if you get an 'epiphany' that helps you figure out what's going on.)
Cut up some carrots into bite-size, non-choke bits. Following what @loosie said, use those carrot chunks as rewards for her coming up to you while you have the halter. You have to present both things to her, and give it time to work. If she wants more carrots, she has to deal with seeing the halter. With patience, she'll stop caring about the halter because it's not doing anything...yet.
Once you've got her following you around for carrots, put a leadrope over her neck while she's chewing. Hold the loop firmly, but keep the loop around her neck a little loose so that it doesn't startle her. Now see if you can gradually introduce a light pressure on the lead rope - you want her to take a step "forward". Be prepared for her to pull back or obey. If she pulls back, don't let go until she 'gives' into the pressure. When she gives, release the pressure and give her a carrot. Then try again until she starts walking around with you. (Since she was haltered and leading before, it shouldn't take long for her to go back to being good about leading.)
As soon as you can lead her with the lead rope, get her into a roundpen or smaller enclosure and do some basic 'stop and go' exercises. Once she routinely stops and stands for you (and you've rewarded her with carrots for doing so), it's time to get accustomed to the halter. Rub that halter all over her face and body until she's licking/chewing, calming down, and lowering her head. Then walk away from her with the halter, and see if she follows you. If she does, give her another pat and a carrot. Then put the halter on, and continue feeding carrots. Then lead her around the roundpen. (This is a best case scenario - let us know if you run into problems.) Then practice taking the halter off and putting it back on. (If she bolts when you take it off, chase her off - she's not allowed to point that butt at you, not here and not in the pasture either. When she comes back, carrot and pat. Repeat exercises.)
To be honest, I'm not a big fan of the 'food' method, but it does seem to work as a preliminary step to getting your horse to enjoy interacting with you. I mean, what teenager doesn't like food? And humans are such good providers of food - just like herd mares who lead everyone to good eats! Good luck and let us know how it goes. There are many methods besides this, and it's good that there are - all horses are different, and we need everyone working together to find a way to communicate.
As for the mounting up issue:
(Use a mounting block. It's easier and safer for you.)
Related helpful videos to build perspective on the situation of "horse worried about (x), won't stand still." https://youtu.be/Gfy-Up5dCvE https://youtu.be/X0UFpYjYHss https://youtu.be/cfuRH383Lio https://youtu.be/MJ5UCavHwxk
(Every trainer is different, and some uses different words that amount to the same thing. Explore Youtube and find a trainer that works for you. Ask for our opinions of the trainer if you like.)