I grew up at a field elevation of about 5,000', and had typical field density altitudes for takeoff in the summer of around 8,500'. It was a mountainous area, with tops above 12,000', so typical flights in light, normally aspirated airplanes were very often above 10,000 MSL.
My best effort in a normally aspirated airplane was just shy of 18,000' in a Cessna 150. The airplane won't climb there on it's own, of course, but using thermal lifting or orographic lifting plus engine power (or what's left of it)...it will.
What had inspired me at the time was an article by Barry Schiff regarding setting up a light airplane with the a high glide ratio. He talked about setting up a light airplane with a low power setting to fly at approximately a 26:1 glide ratio to simulate a sailplane. He then discussed flying the airplane using soaring and thermally techniques and observing the altitude gain and performance. I happened to have a Cessna 150 available to play with, and spent about three and a half hours getting up there, and then back down.
If you're going to undertake that kind of a project, of course, have supplemental oxygen available; this should go without saying. If your'e accustomed to living at sea level, then even a climb to 10,000 might consider taking oxygen. It's not necessary, but your own tolerance depends on your physical condition, age, rest, etc. In the US, the FAA has long recommended oxygen above 10,000 in the daytime, and above 5,000' at night.
Yes, the engine must be leaned; any time a normally aspirated engine is being operated above 3,000' or so density altitude, the engine should be leaned.
There's nothing particularly magic about being at 10,000', but enjoy the trip. If it's not something you do all the time, then any new experience can be a bit of an adventure and enjoyable. Take some time when you get there and examine the airplane performance. Play with the mixture. Try different rates of climb in 5 knot increments, and compare it to the aircraft handbook performance. Pull the power back and try descents at different airspeeds, in five knot increments, note the rates of descent.
We typically flew older normally aspirated Cessna 182's to 15,000 or 16,000 with a full load of skydivers, all day long in the summer, on the weekends. A typical climb to altitude and descent ran about a half hour to 45 minutes, total.
A warrior or cherokee will get up there just fine.