I think it's important to realize that you CAN keep things cheap, if you try. There will always need to be an "emergency" fund, but you can supplement this by keeping costs down in the meantime.
This may sound silly, but when looking to buy your horse, look for an "easy keeper" with good feet. Sucks to limit your buying possibilities, but you WILL save money. For example, I live in a northern climate and my horse is outside 24/7. She is a fat, healthy, 13 year old mare. She goes barefoot year round because she has excellent feet, and is on zero supplements (not even grain, except when I'm working her hard every day). She does just fine on free-choice hay.
I realize that this is not a possibility for many horses, but I also feel that sometimes people tend to pour money into horses for reasons that are not really essential.
24/7 turnout is cheaper, as well as healthier for the horse.
I've been riding for my entire life, and most of my tack has been second hand. I have a very hard time justifying buying new when they are so many great advertising websites to buy from (kijiji, ebay, craigslist), and lots of second hand tack shops around.
Show fees will depend - here, I think dressage classes are about $10 each.
I can't comment on shots because we don't do vaccinations here (located very far north with small horse population). You will need at least a once-yearly coggins test though. Those are quite cheap I believe (expensive here though). We worm our horses ourselves 2x/year for around $10 each time. I've heard you can give some vaccinations yourself, and purchase them online for much cheaper than getting the vet to do it.
And definitely just learn to pull manes/clip yourself, it's really not very hard :)