How can I get Equus in the easiest or fastest way? 
- Enter competitions with your horses
Vote in the Creation Space when you have unlocked it
Breed and sell horses
Sell items you don't need
Another tip: Don't rush things, take your time - it's worth it.
Especially if you "must" have an EC - that is the least of your concerns in the beginning, take your time to establish yourself and get a good foothold on the game
- Raising cows for leather is probably the quickest. Crafting leather in the workshops nets you a bit more but it takes longer and you have to maintain the workshop so up to you if that's worth it to you.
Entering a bunch of competitions is also pretty quick (you get equus just for entering). Divine comps pay the most; and you get 3 free Divines from the Ascent tutorial so working them daily to build up a stock can be worth your while.
Other ways include selling excess items (I don't recommend just net-selling everything since some stuff will be useful in the future, but once you know more about what you will and won't use, you'll have a better idea) selling foals, and offering stallion covers.
One thing to note about dropping sales particularly; you can sell droppings but if you're not done your beginner objectives, you need around 500 in store to complete those, so don't sell them until you're done. After that, keep a stock of about 100 at a time. Daily objectives require between 60 -100 droppings to be sold for passes, so it's worth your while.
- Adding the Parchment Paper bonus to you horses helps with equus.
Adding the Parchment Paper bonus to you horses helps with equus.
This helps a little with equus indeed, but it stops you from getting droppings.
I would not recommend this, because droppings are something you will often need. For (daily) objectives, for trophies, for making 3*** fertilizer in your workshop and for feeding Plant divines once you get any of those. And I can tell you, they need a LOT
Raising cows for leather is certainly most profitable but not the quickest when you are new. They take 30 days.
they take 30 days AND you need 360k eq for all 6 of the 25 acre meadows, not to mention if you want to increase the amount of leather you'd need 239,922eq for the feed troughs. So you'd have to drop pretty much 600k, and wait 30 days to actually get the eq. Definitely not fast at all!!
Adding the Parchment Paper bonus to you horses helps with equus.
You'd get more eq selling the parchment in exchanges for 10k eq than you would caring for your horses all day. And as a new player its extremely important to build up a stock of droppings, like meeki said they're needed for objectives and plant divines if they want them.
Personally, if you had the time and ap you'd make a decent bit by blupping top gp horses in sales and putting up public covers. Depending on breed, top gp may be pretty cheap and cover gp isn't always up to date with sales gp.
