I like to think of stirrups as ABS: It's nice if they are there if you lose lateral balance, but there's also a chance you can lose them on a ride at whatever pace, and you should not feel as though you can't stay on the horse without them.
I ride in an English saddle, mostly on trails, including up and down ravines. Downhill, I don't lean back so much as continue to stay vertical over the horse's center of gravity. As the horse is tilted downwards, yes, relative to the horse's back I'm tilted backwards, but not with respect to gravity. I keep myself from sliding over the horse's neck by pressing the inside of my thighs against the saddle pads, but I keep my pressure on the stirrups light. Stirrups give you zero support to stay in the saddle if the horse starts sliding or speeds up a little towards the bottom of the hill.
As for stirrup length: A bit shorter is easier for posting the trot, a bit longer is easier for cantering. I had an endurance rider tell me that they actually adjust stirrups on the fly (while moving) depending on what they expect to be doing for a while next: One hole longer for a flat canter stretch, one hole shorter for trotting or jumping. You should definitely play around with different options to get a feel for what makes you feel the most confident.
I don't know if you can do that in a Western saddle equipped with a liver puncturing device (a.k.a. horn), but when my horse goes up a hill, I get off his back into two-point, so he can find it easier to step under himself. It's also an excellent strengthening exercise if you do that for several minutes. I do this for all gaits.
If, as you say, you just started lessons, some of the issues you face are not a matter of being shown what to do, but being able to do it by virtue of your core and leg strength. That's something that'll develop more slowly than your understanding of how to do it correctly.
In any case, riding is a biomechanical and physics exercise. Don't listen to what it is supposed to "look like" if it contradicts basic physical principles. In particular, you want to find a body position that requires minimal sustained muscular effort to maintain - that's where you are in balance, and that's where you can relax the most muscle groups to actively stay with the horse as it moves. Your contact with the horse should be as close to your own center of gravity as possible: it's easier to keep a broom steady if you support it at the balance point rather than the end of the handle. So you want to support your body in the saddle with your butt and thighs, not your feet or hands.
Anyway, I'm just shooting from the hip with this...maybe it'll give you something to contemplate.
There's a lot of excellent responses here (and I've not read everything yet), but I thought this one bore repeating - and I've added emphasis to a really really really important thing that not a lot of horse-riding books or instructors will tell you!
Your personal range of good positions on the horse will depend on your horse, your saddle, your style of riding, what you're doing that day, the way you're anatomically put together, your current level of fitness etc etc.
I personally have really long femurs and therefore prefer to ride "long and relaxed" on trails - almost in dressage position. This is also more or less what happens bareback, which is good practice for secure independent riding (maybe do stirrupless first
). Like many tall people, I'm also something of a hunchback, plus I have mild scoliosis, so I do Pilates to help counter this stuff, and give me core strength for better riding and general everyday life.
This is a side view of my leg position on my riding horse Sunsmart, who'd built like a beer barrel:
It's fine for jumps and galloping too, as long as I don't try Olympic jumping. The best position for you is a very personal thing - and you know you've got it when it feels effortless and allows you to move in harmony with your horse, and you do well together, and neither of you is sore afterwards.
And then you can adapt it, and have spontaneous fun - as happened here:
I'll always remember this moment, on my friend's OTTB, in the Albany Harbour. We were doing a photo-shoot of her and her horse, and at the end of it she asked, "Do you want a go?" Her Rikki-Tikki was a wonderful, mad horse, and Sunsmart's riding buddy when we had him agisted, before we moved to our farm, so I couldn't pass up the opportunity to frolic in the water with this lovely, huge, personality-plus horse. Only thing is, her stirrup leathers were too short for me even at full extension, so I had to ride him stirrupless, the first time I rode him! Haha, life is fun. I got soaked like a drowned rat from all the splashing of this horse running through the water on the sandbank beneath. Brett asked me to come up to him for a photo, which we did rather rapidly, and this resulted in a few bits getting cut off the photo and a wonky horizon, but it just so perfectly sums up the ride!