Both strength and bat speed are important when it comes to hitting. Are there routines/exercises that focus specifically on increasing the speed of one’s swing?
I have forwarded your question to my husband, a serious baseball coach. 
And here I am! 
There are three absolutely critical elements to getting a quick stroke: The most important is technique. I’ll get to that last. The other two keys are your hands/wrists and your lower half, both of which are very much tied to the aforementioned technique.
Your lower half is essential for everything you will ever do athletically. The basics are going to help you here – tons and tons of crunches and back extensions to strengthen your midsection, lunges and running or cycling to get your legs good and strong. Power comes from your lower half, and so you need to have strong legs to drive the ball and strong abs to transfer that power upstairs to your hands.
In the upper body, you’re going to want to work on your arms. Pushups never hurt, but try doing them up on your fingertips, to work your wrists and hands more. The key here is wrists – they have to be strong and quick. Squeezing a tennis ball or balling up sheets of newspaper, or repainting your house and doing 3 coats with the paint brush will all help strengthen your wrists.
The other thing you really can’t do enough of is stretching. Staying limber will allow you to be more explosive and quick, which is the whole point.
Now technique – this is where the rubber hits the road. You need to swing your bat 50-100 times a day, preferrably NOT with any extra weight. Extra weight can cause you to drop the barrel and that causes pop-ups and weak ground balls, which is not the goal. Really focus on having your hips lead your hands – take your stride (no more than the width of your foot), explode your hips to the pitcher, until your belt buckle is facing him. Clearing your hips strongly and quickly helps start your hands in motion and reduces the distance they have to travel to get the bat to the ball, which quickens everything up. Next, throw the knob of the bat (not the barrel) at the ball. As you approach the point of contact, your top hand should push the barrel through the hitting zone, aiming to hit the ball on the top half and on the inner half, such that you are trying to drive the ball hard, down and to the opposite field. A great drill for keeping your hands in tight and exploding to the ball is to go up to a fencepost with the bat pointed at the post, until the end of the bat is touching the post and the knob is touching your belly button. From that close in, assume your batting stance and take a full swing, making sure to clear your hips with your hands back, shooting the knob through nice and tight to your body, and then launching the barrel once the bat is clear of the fencepost. I can take a full-speed cut like that both right-handed and left-handed, to demonstrate for my players. You can take tons and tons of dry cuts, with an imaginary pitcher and imaginary ball, just focussing on getting the technique right and repeatable every time, until it’s locked into your muscle memory.
So ultimately, for hitting I’d be looking to work wrists, shoulders, legs and abs to help develop your bat speed, and then just swing the bat over and over and over again. Perfect practice makes perfect!
Hope that helps!
Thanks much! I take it muscle strength does not necessarilly translate into bat speed. True?
Yes and no. If there’s one thing the steroid era taught us, it’s that big, powerful muscles give an athlete an advantage. You will hit the ball farther and harder and you’ll probably have more bat speed. However, bad mechanics trump muscle mass in most cases, aside from the most freakishly elite athletes. If, for example, you pull your had off the ball and can’t track the pitch properly, you’ll never hit a curveball, except by accident, and it doesn’t matter one whit how strong you are.
The other thing is that bulk can work against quickness and flexibility. Darryl Strawberry was one of the first players to jump hip-deep into bulking up in the late 80s, but he didn’t really know what he was doing and, consequently, had a terrible season after building all the extra muscle because he could coil and release properly – the extra muscle, built up in the wrong places, was actually working against him. The advances in baseball-specific training have come by leaps and bounds since then and there are plenty of very specific programs out there to target very specific muscle groups – I have, for example, the Toronto Blue Jays’ pitchers’ shoulder program, which involves a dozen low-weight (5 lbs or less), high rep exercises using free weights and resistance bands. It works like a damn to strengthen the rotator cuff muscles, but does next to nothing to work the big muscles around the shoulder.
So, for me working with mere mortals looking to get better without trying to get to be Albert Pujols-good, I like to keep things simple – pushups, crunches, lunges, squeezing tennis balls constantly, stretch tons to maintain your flexibility, and swing, swing, swing. It’s more about tone than raw power to get through the hitting zone quickly.
Having said that, I can dredge up a couple more very specific exercises if you’d like some more technical stuff to attack particular muscles groups for tone and strength.