Before I got my horse, I had to do a lot of research too. I get free board where my filly is at, but I have to do choirs when they need me. So boarding is not an issue for me. My parents (I am 15) said they would pay for the horse, for my 15th birthday, but I had to pay for farrier, vet(they would help, if I needed them though) all the tack and supplies, grain if I fed it to her, and everything a horse would need.
So far things have gone good. Breeze was 150 dollars, and I have spent around 300 dollars on her already, in 5 months, I have not had to pay any vet bills yet, but... Also I live in Canada, so it is different from the state
Here the farrier costs 20 dollars But in the state it costs more, save around 50 dollars for farrier everytime you need it.
A vet exam costs 70-90 dollars here, just an exam, not sure there, but put away as much as you can for vet bills. There is also wormer, costing 20 dollars a tube, you will need that every fall and spring, maybe more. ( I only worm my horse in fall and spring)
Tack... put away around 1000 dollars for tack, so when you need it, you can buy a good saddle, bridle, saddle pads, reins, bits, etc. You might not need that much, but saddles are expensive.
Supplies... you need to buy grooming stuff like brushes, hoofpicks, etc, shampoo and conditioner if you are wanting to show, blankets if your planning to blanket, including a winter blanket, and fly sheets, or coolers. Shipping boots are useful when hauling your horse, fly masks, all this coming close to around 700 dollars give or take.
For you...the rider... prober footwear like riding boots, if your planning on showing, then nice showing clothes, maybe even riding gloves, depending on what you are going to do... comes to around 500 dollars, boots are expensive.
You do not need to save all the money at once, like me, I just buy what I need when I need it, like right now I am broke, but I have the stuff I need. Since I don't have a job, I am just doing small tasks that get me enough money to put towards my horse. I got smart this way, and got a younger horse so I don't have to pay for tack right away, and I also wanted to train it myself... so it worked out.
If you are wanting to save for everything right away, I would save a minimum of 3000 dollars, not including the price of the horse... you will need a lot of the money for sudden vet bills, you could even start a savings account that goes right towards your horse, and don't use that money for anything other then the horse.
Hope I helped, but I might not be that accurate.