Frances and I take a bottle of whisky with us to the site, the digger man gets a swig, the vet is offered one, but never takes it, we take one or two or three and drink to the passing of a dear friend, it helps us!
Then we spend the rest of the day bawling, everyone gives is that space for the rest of the day then slowly, very slowly you can come to terms with it and start looking at photos and the tack etc. as eml says.
Deal with it as you feel you want to, if you want to bawl, do so, for as long and loud and often as makes you feel better. Don;t let any one tell you haw you should feel, they will never know.
We have a spot in a park where a lot of ours are burried. My old horse is burried at the bottom of the "garden". That's nice in a way, but that's not an option open to a lot of folk these days, cremation is so common now, but you can scatter the ashes in a favourite spot.
When all's said and done, and you have got over the dreadfulness of it all, the most important thing to remember that ONLY caring owners do this for their animals, letting them go when the time is right quickly and with as little suffering as possible is the most important thing. Somtimes it is a decision that is so hard to make, sometimes it is one we have little choice over.
Anyone contemplating owning a horse should think very carefully about this subject as sensetive as it is, because you might find yourself in the situation out of the blue and need to deal with is sensibly, surgery etc. can be an option, but it is more often kinder to let them go rather than grasp as straws.