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I'm not sure if this question is about getting money or horses, but I'll try to answer both, because they are different questions.
Horses
There are two ways you can acquire a horse - buying one or stealing one. Sometimes you may be lucky enough just to find a horse without an owner. For instance, two days ago I was wandering the Ivarstead-Helgen mountain pass and I found a horse on its own. I rode it to Helgen and then left it there since I don't really use horses. You can also steal horses if you see someone riding one, but you may get a bounty that you'll have to pay off later in the hold. There's also a video here on IGN about acquiring horses.
Gold
As for acquiring money, when I was in early game and had very little gold, the most effective method for me was the immoral way of the grave-robber. I would go into barrows, shrines and tombs and would loot every corpse and open every sarcophagus or urn that I could. Some of the urns will give you only 5 gold or so, some 30, but because there are so many it all adds up very quickly. I acquired several hundred gold within a very short space of time.
You can sell weapons. If you go into tombs and kill as many enemies as possible, you can collect their weapons and armour and sell them at forges. They won't sell for much, but if you enchant them, then you'll get more money. Selling jewellery is another great way to make money quickly. Silver and gold sells for quite a bit, more when jewel-encrusted and even more when enchanted. You can smith these items yourself at an anvil/forge. Horses only sell for about 1,000 each, so you should find yourself with enough money very quickly.
If you want to get gold ingots to smith for jewellery, a relatively easy way is to find the Dead Man's Treasure, which is an unmarked location in coldest Northeast Skyrim. Possibly this castaway came from the wreck of the Winter War, which lies south-west. You should find around 6-9 gold ingots there:
https://elderscrolls.fandom.com/wiki/Dead_Man%27s_Treasure
You may have to travel quite a distance to find it, though that depends on where you are and how travelled you are in early game.
You can travel to Kolskeggr Mine in Markarth and deplete the place of gold ore, travel back to Markarth and smelt the ore into ingots, you should be able to get a decent amount of ingots, at least 18 if I remember correctly. Then you can make jewellery out of those and sell that in the city. You may find, especially at lower levels, that making the journey from Whiterun to Markarth to be quite hazardous and the mine itself has a few Forsworn in it with at least one Briarheart, but these aren't that hard to beat. I personally found getting to the mine at level 2 to be harder than dealing with the miners.
Also, potions. Making potions can yield a lot of gold. This is something I haven't personally tried, but there's more on that here and here. If you open the menus within the page in the latter link called "Base Game and Official Add-On Ingredients" and "Base Game Ingredients" it's a great starting point for telling you what you need to find to make potions.
There are loads of articles and threads on the net about how to earn money in early game, but these were the ways I did it.
For instance:
https://skyrimforums.org/sf/threads/making-money-early-in-game.22726/
https://levelskip.com/rpgs/Skyrim-How-to-Make-MoneyGold-Quickly
https://steamcommunity.com/app/72850/discussions/0/451848855000864460/
