If you expect to get long life from a CT-series bike, particularly one used on the road, buy it oil. Change it often. Ignore the interval you use in your car, because 3000 miles is
to a bike this small. There are some CT-90s that live their whole lives and never reach 3000 miles. Therefore, change the oil every 500 miles within a season, and change the oil every year as a minimum.
There are exceptions... if you blow a clutch, change the oil twice. If you trash a set of rings, change the oil twice. If you have a shop do major gearbox work, change the oil when you get the bike back.
What to use?
The oil you put in a CT-90 depends on what you do with it. If you ride only in the summer, and ride lightly, and not often, 10w30 is probably fine. A decent detergent oil will keep the thing clean pretty well. If you ride a lot in the heat of the summer and ride hard, particularly wide-open on the street, 20w50 racing oil is probably a better bet. If you ride in the winter, 5w20 is fine, but don't use that 5w20 into the summer... it's not really good for heat.
Never use any type of engine flush in a CT-90. The system will have too much fluid in it and will overpressurize and trash seals and gaskets. If you must must must disobey this warning and use motor flush, drain out one pint of oil (measure it) and replace it with one pint of flush. Run the bike with the flush in it for no more than five minutes and never ever above idle. Ignore this at your peril.
Changing oil doesn't require any special tools. The drain plug is a 12mm bolt, so grab a 14mm (might be 17... I can't remember) wrench to undo it. See the note below about replacing the oil screen.
Now, the official manuals say that the CT-90s used 0.9 liters of oil. Since in the United States, every quart bottle contains 0.946 liters (one quart) of oil, yes, this means you will overfill the bike very slightly every time you change the oil. Deal with it. Are you really going to put that last 0.046 liters of oil (about three-quarters of an ounce) away in a jar and save it for something? Forget it... use the whole thing.
Special note: the CT-110 is rated for 1.1 liters, so maybe you could buy a CT-90
a CT-110, and between the two of them, use two liters of oil exactly.
Put the bike up on its center stand on a level surface. If for some reason the front wheel is off, balance the bike so that it's roughly level as if the wheel were there. The idea is, the bike will not drain completely if it's not level in all respects.
Put a container under the bike's engine to catch the oil. If you're accurate, you can use a quart oil container (empty) to catch the used oil. Undo the drain plug, undo the filler plug (on the right side of the engine case, near the kick start) and drain the oil out. Let it sit for at least ten minutes... this process goes faster if the engine was warmed up beforehand, but be careful, both the engine and the oil could be hot and may melt the catch container if it's too hot.
Once you're sure it's drained, wipe off the drain plug, looking for any metal specks or anything that looks larger than ordinary dirt, and screw it back in. Every four or five oil changes, you could probably stand to replace the copper washer under the plug. One other thing to look at is the condition of the oil that comes out. It should be dark, but still reasonably smooth-looking. If it comes out with visible grit in it, or if it's milky, find out why. Another, very important thing is, if the oil is very thin, like water, has a gassy smell and you get much more than a quart back, beware. It may be that your carburetor is leaking gas down through the cylinder bore and into the crankcase, and it's watering the oil way down. That's a good way to destroy your rings and bearings, and in extreme cases you could ignite it and basically blow the engine apart. Assuming you recycle oil (you should) do not take such oil to the recycler until you've let it sit, exposed, away from anything hazardous like a source of ignition, for at least a week. That should be enough time to let the gasoline vaporize away and leave the oil behind. Find out what's wrong with your carb soon.
Once the plug is back in, carefully pour the new oil into the filler hole. Let it sit for a couple of minutes, then screw the filler cap/dipstick back in and start the bike up. After a minute of running, stop it, let it sit for a minute, then check the oil level.
What about the oil filter?
You don't have one. The CT-series bikes instead have an oil screen. It's not a disposable item, but it can be cleaned, and should be every other oil change or so. Doing so is not a trivial project due to the location of the screen.
Picture that you're looking at the bike from next to the right handgrip. The screen is reached from inside the right engine side cover, the same one you remove to access the clutch. It slips into a slot just above the drain plug.
To reach it, remove the kick starter, undo all the Phillips screws (using an impact driver) and carefully remove the side cover. Slip the oil screen out of its slot, clean it in Safety-Kleen, WD-40 or kerosene, blow it clean, and slip it back in. Pay attention to anything weird the screen caught, particularly stuff like metal shavings and clutch particles. Occasionally, some really interesting things will show up in the filter.
What about transmission oil?
The CT-series doesn't have separate transmission lubrication. Engine and gearbox share the same quart of oil. This is why you have to change the engine oil after trashing a clutch or having gear work done.
Yes, you can put synthetic oil (Mobil 1, etc.) in a CT-90. Stay with the rated viscosity and do not use this as an excuse to not change the oil as often. One thing about synthetic is that it sometimes will leak through microscopic gaps that natural oil doesn't leak from. If you put synthetic in a 90 that has seen some miles, don't be surprised if you suddenly discover new oil leaks you didn't think you had. Consider this a good excuse to replace the gaskets and seals.