Speed plays a major role in tennis: The earlier The earlier you reach the ball, the more time you get to generate more power and accuracy. Reaching the ball late and out of position is the perfect recipe for unforced errors. Hence, including these elements in your tennis classes and training routine can only boost your tennis game development.

There are different types of speed: sprinting forward is one and it is sometimes required in tennis like when we have to reach a drop shot. But if we take a closer look at how a tennis player moves, we will notice often asymmetrical body movements between the upper and the lower parts.
The upper body is facing forward only when ball is in opponent's court and almost always sideways otherwise. If we take a look at the lower body, we notice that the legs are not always running, we do shuffle a lot, we slide (on clay courts), we jump, we make abrupt changes of direction, we do take often very small and quick steps to adjust our position prior to hitting the ball.
Speed for tennis must be developed with these facts in mind. Improving your personal best on a 100m sprint course will certainly help but that's not what is going to make you faster on a tennis court. That is why you need both speed and agility in tennis. Agility means here the ability to quickly shift direction and to do all sorts of other footwork patterns seen repetitively in a tennis match.

Tennis is very dynamic in nature, the court is not very large compared to how fast a tennis ball can travel back and forth from one player to another. Tennis players must constantly be on their toes, in its literal sense! this means heels off the ground, ready to generate a split step prior to moving to the ball.
A split step is a little hop players do to generate momentum off their legs so that they can propel themselves faster toward the incoming ball. a split step is typically performed just before opponent makes contact with the ball. Despite how important the split step is in tennis, many recreational players do not use it and many think it's not really necessary or feel it sucks energy out more than anything else.
The split step in tennis is actually the corner stone of the game and the lack of it will have unavoidable repercussions on the level at which one can play.

The best way to practice the split step is with a jump rope. Here is a cool video featuring a few drills that can be done as warm up before your actual tennis class or training session.

As stated above, a tennis player is not a sprinter, he or she should be quick moving in any direction and able to change directions swiftly and efficiently.
Here are some good examples on how to work on speed and agility to develop this ability.

This is a good drill to improve the split step and subsequent reaction to the incoming ball.
The Split step is a little hop that should be made right before your opponent strikes the ball. It is meant to help you generate momentum, which you can use to spring towards the ball. Think of it like a an imaginary springboard.
How high one should jump when split stepping depends on how much momentum they'd d like to generate. For example, if you are in a very defensive position where you know you could have less time to react, jumping a bit higher when split stepping can catapult you more effectively towards the ball. Similarly, if the rally is more or less even with neither players having the upper hand, a normal split step would be sufficient.

Lateral shuffling is very handy in tennis as it makes the player equally ready to react to the ball in any direction especially when the opponent is about to hit ball. The last thing you want to do is to get caught running in one direction while the ball is flying towards another. Lateral shuffling should almost always be followed by a split step just before opponent makes contact with the ball.

Running is faster than shuffling and can be used to cover larger distances quicker. Though, if your opponent hits the ball in the opposite direction, you will have a hard time adjusting your course. Hence, running is used only when you still have a few moments before a shot from your opponent becomes imminent. In this case, switching to lateral shuffling is required to preserve your balance and ability to move in any direction. Lateral shuffling also goes hand in hand with the subsequent split step that occurs right before your opponent makes contact with the ball.

The following drills combine many directions at once, so it's very close to real match situations. Many variations are possible.