Prerequisites

There are two prerequisites to getting your Lotro Mount

you need to be level 35
you need 4 gold and 220 silver

I highly recommend to get another 100 silver to pay for renting horses. The quest you need to do needs a lot of traveling around, and you don't want to do that by foot.

Then what?

1. Fresh Steed for Bree

In the North of Breeland there is a vendor named Eogar at a  LOTRO Gold Horse farm. You can easily find the farm looking at the map of Bree. It is straight North of the second letter 'N' in the word 'Northern (Bree-Land).

Talk to Eogar to receive the first quest. He asks you to ride a horse to North Bree. Accept the quest, right-click a blonde horse / pony and ride to the stablemaster in North Bree.

1.b. Getting back to Eogar

Talk to the stablemaster in North Bree and buy a ride to Trestlebridge. Jump of the horse as soon as you get close to the farm.

2. Fresh Steed for Michel Delving

Talk to Eogar and accept the quest. Get a horse and ride to the LOTRO Gold Michel Delving Stablemaster

2.b. Getting back to Eogar (again)

Talk to the Stablemaster in Michel Devling and buy a ride to North Bree. Once in North Bree, talk to the stablemaster there, buy a ride to Trestlebridge, and again jump off near the Horse farm.

3. Fresh Steed for Othrikar

Talk to Eogar again and accept the quest. This one is the most tedious part of the quest. The ride to the Othrikar stablemaster is going to take you a good 5-8 minutes.

3.b Getting back to Eogar

Buy a ride to Esteldin, and from there back to North Bree.

4. Proving your Quality

Talk to Eogar again. Before he is willing to sell you a horse you need to prove your riding skills. Accept the quest and race the track around the farm. Make sure to take as many short-cuts as possible (maybe run the track by foot once before you do it in earnest) and the whole thing should be easily be achieved in the given time frame.

5. Done

Talk to Eogar again and you will receive the trait 'Riding'. All you need now is a horse. Buy LOTRO Gold one of the color you like, but make sure you pick the right size for your race (Dwarves and Hobbits ride ponies, Humans and Elves ride horses)