Well it depends how you want to shape the tree. Opened centered, modified leader, or central leader. The first two are best for peaches, but any style can be used. How big do you want the tree? It's hard to give advice without knowing desired shape, and size. My first impression is you have narrow crotch angles and branch a and b seem to be competing for central leader status. No it's a perfect time to prune. You can prune it in spring, summer, and fall. If you want to keep it small, all three pruning times are needed. Again a and b seem to be competing, you need to remove one, or train it down with weights to improve angle. The branches growing almost 90 degrees to main trunk are excellent. Also if it fruits, only allow one fruit per 6 inches, else the branch may break. Remove them when small. The remaining peaches will grow large and taste great! Wait for more advice before you do anything. I just may be seeing it wrong.

Thanks for your reply! I was thinking of cutting branch B all the way down to where that first branch is coming off. Would that be okay?

Yes, I would cut off branch B and leave A as the modified central leader. Like a hair cut it will grow back. I would leave one bud, and not cut it completely off. Always leave the collar when removing all of branch, so it can heal correctly. I think you should leave one bud node, and not to the collar, as a branch might grow in a better direction from that bud. You can always cut it off again. In general, you should trim all last year's growth by 1/3. Even up to 50% if you want a small tree. yes, you are cutting off fruiting wood, but we want to establish the tree first, the structure, and the roots, worry about fruit later. In the long run the tree will produce better doing this. You want 3 or 4 main scaffold branches. nice if not exactly across from one another but spindling up. Trim these lateral scaffold branches to create secondary branches on them that will be where the fruit will be produced once larger. If you are looking from above the plant you want your scaffold branches to be say at 12 O'clock, 3 O'clock, 6 O'clock, and 9 O'clock. (this is ideal, hardly ever happens that well!). Hopefully these will be spiraling up the central leader, and not directly across, but that is OK. If a branch on the central leader is directly above another branch the top branch will shade the bottom. Remove the top one. The top grows more aggressively, once lower wood is gone, it's gone! So I always leave the lower branch. Always try to prune just above an outward facing bud, and on a slant (top of slant closest to bud), leave a little stem above the bud, 1/4 inch or so. Not directly above the bud. You did well removing crossing branches and ones going inward, but I would trim all branches on a young tree about 1/3 of last year's growth should be removed. I myself want small trees so I remove 50%, sometimes more, and almost all of it once it reaches the height I want. I'm growing mine with an open center. Like a bowl.
Also remove secondary branches on the scaffold branches that are growing straight up, or straight down. You want them growing parallel to the ground, or close! Hopefully others will respond too, I'm not like super experienced, but I think what I'm saying is correct.

Thanks a lot for your insight! I made some cuts to the tree, but I didn't have my phone on me at the time so I was unable to take a picture. I will take some pictures tomorrow morning.

I cut branch B about 2/3 of the way down, right above two branches that were growing outwards.I may get rid of one of them. If I do this, then will this branch turn into the new scaffold?

"I cut branch B about 2/3 of the way down, right above two branches that were growing outwards.I may get rid of one of them. If I do this, then will this branch turn into the new scaffold?"

Yes, if it grows in a good direction. Learn from it, leave it see what it does. Two concerns with branch B, one it is competing as central leader, and the narrow crotch angle. So the angle is still of concern.It may be OK, hard to see without being there! It may be OK, I need to look at branch B from the left or the right of current photos to see how angle is. Branch C is in "perfect" position! I guess when a branch is more vertical than parallel to the ground, all the pressure of the fruit is at the "V.", The branch is probably too big to train now. A 60 degree angle is what you want. When you get a new branch that is small and it has a bad angle you can use a clothes pin between central leader and branch to force it down. Weights can be used too, but they can swing in the wind and damage branch. Here is a good article about training
http://learningstore.uwex.edu/assets/pdfs/A1959.pdf

Thanks a lot for your help, Drew. After reviewing some videos this I've done the following:

{{gwi:70139}}

Where you see the dashed red lines, thats where the cut has been made on branch B. The branches labeled in Blue B1, B2, B3, and B4 are the secondary branches coming off of B

My thoughts are to cut B further down to B3 so that gets rid of B1. Cut B2 all the way back to the B. So off of B, that leaves B3 at the 6 o clock position and B4 at the 12 oclock position (so C will be at A will be at 9, and C will be at 3). I think I will further get rid of branches D and E.

Here are more pictures:

{{gwi:70140}}

Again, i will cut B1, B2, D, and E.

Here's another view:

{{gwi:70141}}

So essentially, I will have A, B3, B4, and C as the only branches left. If I were to do this, then wil B4 and B3 take over as the main scaffold branches and give off more branches? What do you guys think?

And will I be cutting off too much?

Thank you.

Wow, looking at different angles, I can see a lot more. I think I would leave it as is, as B1 and B2 seem to be the only branches in that direction. Giving a balance, as that side seems open. Branch A is sort of in that direction. It is not an ideal shape, and further adjustments can be made later. If you do decide to make the cuts, it's not to much, but try and fill open areas, don't create them.
So eventually those branches will be huge, and such. I like the way it looks now with branches in every direction. It's ok to have 5 or 6 scaffolds too! You want at least 4, but 6 is good too! The more the better! You can see it better than I can, but if branch is alone in one direction, and it will cause an opening not covered by another branch, leave it.From here it looks like if you remove B1 and/or B2 it leaves an opening in that direction with nothing to replace it. If that is the case, certainly leave it. Balance the tree in all directions. More branches will come out of B too, wait to see if they would work better. Cutting it stimulates growth.
Good luck with it! My peach tree is still dormant. It currently has 4 scaffolds. No central leader. It was planted late last fall, and was chopped down to only 15 inches! The scaffolds are at about 10 inches, it is going to be kept very small, maybe 8 feet tall. Usually you do not want to cut below 15 inches, anywhere from 15 inches to 4 feet, you can start pruning it back. Since it had scaffold branches at 10 inches, I did cut it all the way down to 15 inches tall, and it looks funny as hell, like a miniature tree (it is not!). I have a nectarine I cut down to 19 inches, it is a stick, no scaffolds at present. It may not make it, but if it does grow, it will be small. These techniques are BYOC (backyard orchard culture) techniques to keep trees as small as 5 feet if you wish. I'm shooting for about 8 foot trees. I will cut off all growth past 8 feet every year. I probably will trim them down lower at times to make new fruiting wood.

Thanks a lot! I guess I'll just leave it as it is for now. I'll let it only fruit 4-5 peaches this year, but let it grow for the most part. And then next year, I guess I can modify the tree a bit more if I need to remove some of the scaffolds.