Double and triple tonging are very handy when you need to play fast staccato especially for repeated notes and it's not that difficult to learn.
When you single tongue, you attack each note separately making a "tah" attack. With double tonguing, you will alternate between "tah" and "kah" attacks to play faster. It takes a bit of time to get used to produce the "kah" in the back of your mouth while playing. You can practice scales just on "kah" attacks to strengthen your muscles back there. In the beginning, your tongue will get tired quickly but it should improve with time.
First practice with only 3 notes. Repeat "takata.... takata..." many times. Then add more notes to the sequence until you can do "taka taka taka taka..." non stop comfortably.
For triplets, use triple tonguing. Instead of doing TKTKTK, you make groups of TKT and repeat as much as needed. You can also do TTK for your triplets. I personally prefer the first method.
Before you practice on your instrument, practice "speaking" your double or triple tonguing.
Say "taka taka taka taka ta...." for double and "takata takata takata...." as fast as possible for triple. This way you can get used to it and add some speed before you play it.
Here is a little video about double and triple tonguing to give you an idea. There are more videos with slightly different opinions about it. Some people argue about "takata" vs "tukutu" or even "tagataga..." I think it's a minor detail. You can try different options and see what works the best for you.
If your attacks are a little unfocused and you're getting a "trrr" sound instead of a nice "tah" when you're playing fast staccato and tonguing in different places in your mouth or on your teeth doesn't really help, you might want to try a more global approach to the problem.
My own personal theory is that the roughness in your attacks occurs because your sound is unfocused for a small fraction of a second in the beginning of each note. We very often forget about sound quality when we are playing fast notes. It is relatively easy improve your attacks if you improve your tone.
First play each note separately with your best sound. Do long tones with each note if you want. Then gradually play the "trouble passage" up to the desired speed. Always try to reproduce the sound you were making when you were playing the notes separately. Improving the tone quality for the whole note will automatically improve the tone quality for the beginning of the note thus improving the quality of the attack.
Although this might not entirely clean up your staccato, you should be quite happy with your attacks this way and it won't affect the fluidity of your playing.
I am associate principal horn of the Singapore Symphony Orchestra and the author of the progressive methods. I'm happy to share my experience as a horn player and teacher with you.