Auto transmission not engaging gear at times

Check the fluid level first. It sounds like a pump pressure problem or the torque converter, which might be intermitantly not locking up becasue of lack of pump pressure.
So, before you put fluid in check level and see if it is on stick at all and what it looks like and smells like. If it smells burnt or looks burnt, the intenal clutch packs may be bad or shift solenoid.
Check the fluid level when at operating temperature and put it into all gears first then check with motor running. Also note how much fluid it takes to bring the level up and do it slowly and check it a lot as you add fluid as overfilling it will damage seals.

There is not much of a way to test pump pressure at least I don't have any test if there is a pressure port or something, it is not given to me either.
Automatic tansmissions are so complicated that you would need to remove it and put it on a bench to work on it. The first thing you have to remove is called, "valve body" and it literally looks like a brain with toms of passages. In there are tiny springs and bals that will shot out and get lost along with the setting if you loosen the wrong thing and even with a manual it is hard to tell which screws or bolts retain the body as opposed to being adjustments. Then inside it you generally need special tools to work on it. They are a different world from manual transmisions which I am more comfortable with. You also have to deal with the differential which are the drive gears. This is what you would find in the rear axle pumpkin of rear wheel drive cars. So, it can get to be a crazy mess with front wheel drive stuff. I would reccomend going to a transmission shop. You will notice that automatic transmissions are the only car part that can support a ton of specialty shops like AAMCO transmissions. No other part is so complicated that it needs a shop of its own and Techs specializing in working on them. So, make sure you go to a Transmission Repair shop. It should cost approximately $1500 to get the transmission done. I forgot the mention that it might be a waste to do just the pump as any damage done by this issue will cause other problems and your work will go for not. So, it is best to rebuild the whole thing. You can try replacing the fluid and if it has a replaceable filter replace it at same time. If the fluid is over 30,000 miles old it will cause a lot of problems. Changin it can totally chagne the tannys behaviour. So, it is worth a try.
Let me knwo how it goes.