Start with a relatively simple small oxer. Crosspole at the front, vertical at the back. Make it fairly narrow to begin with, especially if you or your horse haven't jumped an oxer before. Make it as inviting as you possibly can, to begin with, to help YOUR confidence, because oxers do feel different to verticals. A related line is a beautiful thing when you're trying new kinds of jump, especially if you only have two or three strides between your fences. A grid is better if you have enough standards and poles, but a related line will do if not.
With oxers, because they are wider, your horse does need to be in a good spot or you will take rails. Smaller oxers aren't such a problem but once the size starts getting up there, the takeoff spot is very important.
Be prepared to feel like the jump is bigger than it looks, or you might find the opposite, that it looks enormous to you but isn't actually that big! I have an oxer set up in any course I make, and mine is quite wide so that I don't have to move the standards as I move up the heights, but that's because my horse is a really experienced jumper and I've done a lot of oxers with him. You should start with a relatively small oxer. You are training up to 2'8" so I would recommend an oxer around 2' high at the back, with the crosspole being no higher at the ends than the back pole - just so that you know for sure that you and your horse can 1000000% jump that fence no problem. Oxers in shows are usually as wide as they are tall, or a little wider.
A small oxer is likely to feel the same as a vertical of the same height, unless the horse over-jumps it. It's only the larger oxers that feel enormous!
As for more scary/challenging fences, try taking a few fences from tricky angles, not necessarily straight on. My horse will jump from a very difficult angle and is very quick in the jump-off. The best showjumpers all jump their fences from difficult angles in jump-offs to reduce the amount of time they take. Otherwise, you can hang a tarp over your fence and hold it down with a spare pole so it doesn't flap all over the place. Or a rug. A tarp underneath a fence (especially oxers, once you get good at those) will simulate a liverpool, which you usually won't see until you get up to 3'+ but it's good to train them over smaller fences too.
Traffic cones make great fill, and though you won't come across it in shows, danger tape or flappy bunting makes your fence more challenging to the horse, especially if it's windy. A tarp over the jump with a "sprinkler hose" (you know, the ones with the little holes along them that spray water) at the base makes a great trust-building fence.
Whatever fence you point your horse at, he should be expected to try it from any spot, so if he's feeling reluctant, lots of leg, don't worry about form, and make sure your release is more than enough. Grab mane if you have to, just whatever you do, DO NOT jab his mouth. If you don't have a mane to grab (I don't, my horse's mane is roached) a spare stirrup leather makes a great neckstrap, which you can put on his neck in the place you want to aim your release to for a guide, you don't have to just use it if you're left behind. Just don't grab it unless you have to.
A final note, on your instructor. It's actually not a bad thing to show lower than you train. It's the best policy, so that you KNOW that whatever the course designer throws at you, you and your horse can handle it. I'm training up to 3'4" and still haven't shown higher than 2'6" - that will change sooner rather than later, but the highest I am willing to show with Monty at the moment is 3' and even that is a stretch. I'm training nearly a foot higher than I'm showing... nearly two feet if I go eventing because solid fences scare me so I'd rather that the height was nothing. My horse will blow straight over anything small, even if I'm not committed, but anything above 3' he demands a committed rider who doesn't just drop him a few strides before the fence.
A CLEAN SLATE FOR THE FUTURE