Do you need different tack for hunter and for jumper?
Not necessarily. Hunters have very stringent rules for tack while jumpers allow pretty much all tack that is humane, so if you outfit your horse for the hunter ring, that tack will also be allowed in the jumper ring (except for the standing martingale, which is not allowed for the bigger jumper classes).
How do I read a horse striding chart?
For reference, we'll use the chart provided here. In this chart there are options given for different jump heights, stride lengths and pony heights. Choose the number of strides that you want (given here in rows) and use the corresponding number in whichever category you are interested in. For horses, a 12' stride is common. This means that for a 4-stride line, you would select 60 feet, which is four times 12 feet, plus another 12-foot stride for the combined landing and take-off distances. In the hunter ring, longer stride lengths are usually used in the course design when the jumps are bigger and when the lines are coming towards home.
How do you know when to compete at higher levels on your horse?
This is something that is best discussed with your coach. You should always school over higher fences at home than you will see at the horse shows. This is because the show ring is a much more stressful atmosphere and the technicality of the course can make even small jumps quite difficult. Exercises at home are usually designed to suit the horse, while course design at shows is meant to challenge horse and rider. If you and your horse are competing confidently and comfortably at a certain height, you could consider moving up if you have been schooling above the next height bracket at home.
How to memorize hunter courses and striding
Honestly, it's just practice. If you are tired or stressed out, you might find it more difficult to remember the striding. If you are worried about remembering the numbers, try to look at the course at the beginning of the day, well ahead of your class. If you can, watch others ride over the same course so that you can get used to seeing each line ridden in a certain number of strides. As far as the course itself goes, many hunter courses flow quite well if you know the first jump and the last jump (your options are often limited because going off course would involve taking a line backwards!). Many pros will find their way around the ring with just this information. For some riders, memorizing the colours of the jumps/lines helps to cement it in the mind, while for others it's more a matter of "inside-outside-diagonal-outside-diagonal". If you're really worried, check to see if it would be possible to go off-course without jumping a line backwards. If you couldn't, then you have one less thing to worry about (you would never jump an oxer backwards, would you?). If you're worried about forgetting where the course ends, ask your coach or friends to start clapping as you come over the last line of the course.
Is it acceptable to keep a horse's mane long for hunter classes?
Not unless you are attending a breed show and the long mane is part of your breed standard. Hunters are very tradition-based, and the accepted way is to pull the mane to a length of around 4-5 inches and braid it in the hunter style. There is a reason why you see hunters and jumpers with shorter manes; it's very difficult to release properly over a jump if there is long hair to get tangled in. If you are attending a casual schooling show, you can probably get away with the longer mane, but I would recommend doing a running braid to keep the hair neat and out of the way.