You'll find at higher levels, you need to fly the book speed accurately to get any climb out of her. Also once in the cruise, note the difference in IAS and TAS (use the little white adjustable ASI scale to compensation for altitude/temp). You learn about this in the PPL but until you get to 10,000 feet and see the speed indicating 90 knots when you are used to it reading 110kts that you realise that the books were right.
Don't forget to set 1013.25 on the altimeter as you pass 3,000 feet in the climb and report your height in Flight Levels.
If you want to get down quicker
I don't recommend this either. Apart from the risk of shock cooling the engine, it's not good airmanship to descend quickly as it hurts your ears. I usually descend from high levels at just 500fpm in unpressurised aircraft and the technique I use is to pitch forward for -500fpm, let the speed build up towards the yellow arc then throttle back so you've got some power on all the way down to avoid shock cooling.
Adjust your speed for smooth/rough air accordingly.
The distance to the horizon is 1.25 times the sq root of the height in feet so at FL100 you should be able to see 125 miles (vis allowing). Also at that height you'll almost always be above any inversion and the general haze of lower levels and in a standard atmosphere, it will be 20 degrees colder up there too, so wear a woolly!
While doing my PPL I was briefly taught leaning but have never tried it in practice
I had the same experience in my PPL and frankly it's shocking behaviour on the part of the schools. With the high cost of fuel, it's negligent to leave the mixture on full and it's not too kind to the engine at high level. How anyone can fly a plane and not use a primary engine control lever beats me. It's not your fault, but I do suggest reading up on the technique both with/without CHT/EGT gauges. Even an approx inch or so is much better than leaving it all the way rich.
Finally in the long descent at low power settings, even clear of clouds, carb ice is a real potential issue, so make sure you use carb heat according to the book.