Hello Mrs Dinz
As addict says, this is not going to happen overnight. Yes, clear away the turf for a space of about 3 feet all round the trees and put some nice compost onto the soil there instead but keep it away from the trunk by a few inches. That will let the trees get access to nutrients, rather than let the grass get it all.
The pruning is going to have to be done gradually over a period of about three years. The danger of pruning a fruit tree all at once is that you shock it too much and it sends out what are called water shoots. These are great long leggy useless things that just make the problem worse.
In November, when the leaves have fallen and you can see the shape of the trees, you will be able to decide which branches need to be taken out with the greatest urgency. These will be ones that are completely dead and ones that are crossing over the centre of the tree and clogging it up. The idea behind pruning is that you open out the centre of the tree into a nice cup shape. Doing this this winter will probably be enough for now. Next summer, you will be able to have another small snip at the growth made in 2014, just trimming back the side shoots to 5 leaves. In winter 2014, you will be able to have a second major hack at the big branches, making the tree more cup shaped again.
Bear in mind that shoots (not branches, just small shoots) that grow vertically are generally fairly useless for making blossom and should be the ones pruned first. Ones that grow more outwards are better for fruit and should be left to the end when you can see the shape forming.
The most important thing to make a note of now is this: are the fruits you can see now growing on the two trees at the tips of the branches, along the length of the branches on little spurs, or both? Making a note of that will be of great help when it comes to pruning them and also identifying which variety they are. The first lot are known as tip-bearers, the second as spur-bearers and the third as partial tip-bearers and you have to know this before you start chopping bits off otherwise you will maybe end up with no fruit for a year.
Judging by the two photos, you have a cooker and an eater/dual purpose apple. Their shapes, one quite upright and the other more spreading/drooping are quite natural and will happen whatever you do with the pruning knife.
Good luck and have fun.