For every handful of 'greens' add a handful of 'browns'.
Greens are fresh stuff - so like the stuff you have already listed.
Browns are 'old/dead' plant material - so cardboard, shredded paper, straw, etc.
As the greens rot down, if there are no browns, you just get get slimy stuff.
The best way to do it is to layer it browns/greens/browns/greens throughout.
If you have used weed and feed, don't add your lawn mowings as when you come to use the compost, it can stop the plants that are in it from taking up nutrients. You should check that the horses haven't been fed any grass/hay/straw etc that was treated with a ragwort killing herbicide, as this has the same and much stronger effect.
The smaller the bits that go in, the quicker the compost.
The more you turn it, the quicker the compost.
I put all my weedings onto the grass, leave it for a day or two to die off and then mow them and add them to the composters.
I compost where I am needing the compost, then lift the compost bin, move it 2 feet away, and pitch the uncomposted material into the empty bin; and the composted stuff I just rake over the soil and let the worms do the rest. It saves alot of wheelbarrowing stuff about the place.