So playing long tones strengthens the muscle and improve the sound right? But how do I practice longn tones? Use scales? And for how long each time?

when doing long tone scales, i play each note as a minum at approx. 90bpm
it is also good to practice playing a note for as long as you can, such as for 20 secs, more if you can. using different notes is also useful. when playing this note, i start in pp, work to ff, then to pp again, before fading out~ that really does help.
if you just do that for 5/10 mins at the start of every practice, it should be all good

Good question. A million answers!

Register slurs are great(let me know if you aren't familiar with these),

"24-count" excercises are good, where you hold a note for 24 beats .. crescendo for 12 beats then decresc. to nothing for the last 12 ...

Klosé mechanisms are great starters for long tones that use fingers. The Klosé scales can be a "long tone" if you're really advanced.

When I started long tones, I was using scales or playing in fourths or fifths to cover the first two registers.

Lately, I have been using this long tone exercise that uses the 12-tone row:

www.timpricejazz.com/lessons/longtones.pdf

When my teacher saw this, he suggested I use dynamics and a tuner as well.

Yea hump does register slurs mean stuff like low C to G, low G to D and stuff like that?

Yep, and you can jump to the altissimo register once you get to Low-A. "low-A...E...high-C# etc by picking up (or rolling down) the left hand first finger

Another thing that might be helpful is to practice with breathing excercises. Like breathe in for four seconds, then breathe out for four seconds. Or you could breathe in for two seconds, then out for eight. Ya know, switch it up a bit. It's helped me with my breathing while I play.

My band teacher at my school has a basic warm-up for us, if u want warm-ups. We breath in for 4 out for 8 2 times, then 4, to 4, then 4, to 14, then 1, 4 or something like that. After that we will do F stretches (we havent learned lip flex) (up and down) Then we'll sing an F, humm it, play it and sing it 1 more time. Then we'll play 1 of 4 scales (our whole band class hasnt learned more then 4, E flat F, B flat, and A flaT. Then we will play a chord to practice balance (if ur warming up by urslef, no point in doing chord. I'm sure u could implement the long tone in there. BTW, we do Quarter, 8th 8th 8th 8th 8th 8th Quarter for scales.