I'm currently trying to learn scales, arpeggios, and thirds.

For reference, I'm working on G major, Bb major, and D major, on the second page of scales at the end of the 2nd Standard of Excellence book.

Trouble is this: I play through one set of scales (e.g. G) until I get it down and it feels right. Then I move on to the next set (e.g. Bb). Once I have this down, if I go back to the previous scale I was working on, I've forgotten all the hard work I put into it and start missing notes, mostly the ones that are specific to the scale. I might play a Bb instead of a B, or an F instead of an F#.

Does anyone have any tips for getting these things down?

How much music theory do you know? Knowing the makeup of the scales will help you remember whether notes are flat, sharp or natural.

Getting the fingerings in the chromatic scale down is also important.

kornork, it seems like you are practicing the right stuff and are being diligent with what you are working on. It just takes time to feel comfortable in all keys. Keep up the good work and be patient- it will come :)

I play the chromatic scale slowly every day as part of my warm up, so I know the fingerings. I don't know much about music theory, but I can always hear when I've played the wrong note immediately.

Good players practice till they get it right; great players practice until they can't get it wrong. Someone said something like that... not sure who off the top of my head, I'd have to look it up...
There are no magic formulas, but you can come at it from a lot of different angles. Play them fast, play them slow, play them backwards and forwards... it's hard to get them down at first, but don't get discouraged. You got this.

"An amateur practices something until he gets it right. A professional practices it until he can't do it wrong." -Stephen Hillier

If you're making mistakes, you're probably going too fast. Practice at a tempo where you play it right on purpose, not by accident. Remember that it takes time and daily practice to really learn anything.

Developing muscle memory of all keys and fingering combinations takes time- Pedal C is right in saying that practicing at slower tempos helps to get things under your fingers and not practicing mistakes.

Here's a couple of things that might help:
1)While you're watching TV, set and work the finger patterns. Get use to running up and down the scale without blowing into the horn.
2) Once you practice a scale, play four or five simple songs in that key, Mary Had a Little Lamb, Silent Night, etc...
3) Learn your sharps and flats. If someone says Key of D you automatically need to know F# and C#. Using flash cards for this can help.

I sencond that knowing the theory/makeup of scale is important.

just note that the order of practicing G -> Bb -> D is reason of your forgetting and confusing.
I'd suggest the order to increase/decrease # or b by one.
C -> G -> D ( -> A -> E -> B -> F# -> C#)
C -> F -> Bb (-> Eb -> Ab - Db)

play simple melodies on each learned key, before moving to another.

once you learned then, Bb - F - C -G -D, try also moving the key one note up/down,
Bb - C - D, F -G (- A)

once you learned all major scales, try also moving the key chromatically up/down. etc.