Cut the central leader where you want the next layer of side branches to be. Then cut back the side branches you have back to about two feet or so. I would also cut some of the lower branches off at the main leader. Cut all the lower branches off below the three side branches that are near each other. Then cut the three side branches back to about two feet. That's all there is to it, trees are very forgiving, so do not worry about getting it wrong. There is no one way to prune, I prune some with a central leader and other with an open center, that is apple trees. Prune to invigorate the tree and keep in mind that the tree stays the size you want it to be.

What's the purpose of cutting side branches back? Will the tree not then be inclined to send out overly vigorous and upright growth from the cut end?

How it works is you come back and make new cuts every year. So reactions to previous cuts not wanted are dealt with before there has been years of a branch growing in the wrong direction and so forth.

This is the kind of thing where to have a satisfactory outcome you may really need to find out how to go about it. There are actually several styles of pruning of apple trees, with there not being just one way to always go about it - if you did a lot of looking on the internet you probably encountered examples of these different styles being identified and shown. Ideal pruning regimes also vary somewhat with individual cultivars and how they grow and bear.

If you looked at multiple web pages and didn't get anything out of it then you may have to try some other aids such as in-person classes or workshops, pruning videos or even hiring someone to come and help you on the planting site.

I don't know, that looks awfully spiny; maybe it's an ungrafted rootstock like M111?

What variety is it? I do not treat various varieties in the same manner. Most often the best approach is to remove all branches more than a third the diameter (where it joins the trunk) of the trunk. Leave all other branches until the tree begins to fruit and then begin choosing your permanent scaffolds and thinning out the tree. Also keep the tree at the desired height by cutting back to a small branch when it exceeds it.

As a general rule, the less you prune, the more quickly fruit trees come to size and begin fruiting.

I only cut back branches with varieties that tend towards lanky growth and are reluctant to send out secondary branches from scaffolds (main branches connected to trunk). If these cuts are needed your best response will come from waiting till shortly after first growth in spring to make the cuts and coming back a month later to cut back competing leaders at the point of the cut. Otherwise a stub cut can deform branch growth.

You need to eliminate the competing turf below the tree if you want the tree to grow up to its potential in a reasonably short time.