When a wind instrumentalist slides from one note to the next, they refer to the action as "slurring". For trombonists, however, the action is called legato. The reason is because a trumpet player can slur by simply blowing and moving the valves, whereas a trombonist needs to make use of a light tonguing motion known as legato tonguing, in order to avoid "smearing" into each note played.
The terms "slur" and "legato" often mean the same thing to trombonists. Both actions involve playing two or more notes with a smear-free, flawless connection. When it comes to performing slurs, trombone players have various options to choose from.
At one point or another, most trombonists will have difficulty performing a clean, singing legato. Novice players have the most trouble, due largely in part to the majority of method books introducing slurs early on before the young player develops the coordination needed to achieve seamless results.
A player can slur on the trombone in five possible ways:
In everyday playing, the first three types of slur mentioned are the most significant. However, they are best learned one after the other. That way, a beginner can see their progression and build upon a skill that was previously learned.
The first step is air. From the start of the slur to the very end, great legato requires smooth, plentiful, and constant air supply. Great lip slur practice can be found in the "Harmonic Series and Flexibility Studies" from Emory Remington's Warmup and Daily Routine.
To help you envision smooth air, try buzzing the trombone mouthpiece. When lip slurs start to feel and sound right, your next step is a steady, quick motion. (Tonguing isn't permitted as of yet!) If you practice cross grain slurs, your technique will get better. Remember to keep the air moving, except now, you're adding the slide. Also, keep in mind that it's extremely hard to play smooth lines with a damaged slide, so make sure that your slide is aligned and in a dent-free area.
Avoid holding your horn with the right hand. Your right hand is for sliding; your left hand is for holding the horn. The slide should move just as quickly from 1st to 6th position as it does from 1st to 2nd. Once you master these with enough practice, it's time to move on to tongued slurs.
Tongued slurs are the combination of air from lip slurs, the quick slide from cross-grain slurs, and a little bit of tongue use. This may seem like a lot of things to do at once, but with enough practice, you won't even notice all the air and sliding, and you can concentrate on what to do with your tongue. The string of actions are as follows:
The first note is played, then it's time to play the next note. Your tongue rises to put a dent in the stream of air, the slide is moved and/or your lip is changed for the next note, then your tongue drops out of the way. If you got the timing right, you've slurred. Make sure the tongue stroke isn't too heavy. A syllable like "dah" or "rah" will do. Also, never stop the air flow. You're only denting it.
Tongue with your tongue only, not your jaw. Certain performers may rather tongue on cross grain slurs to make them match the other slurs, but either way is fine. Continue buzzing and singing as to not forget about the air flow. There are many great etude books available for improving on legato. Try and find ones by Alan Raph, Reginald Fink, and Johannes Rochut.
It's important for legato to be part of your daily routine. This includes working with either a school music teacher or a trombone teacher on slurring. When a live person is providing you with personal feedback on your playing, that's when real improvement truly shows. Also, remember that nothing compares to studying privately with a renowned teacher or player. In fact, even just one lesson a week will speed up your improvement remarkably.