Did you play at a high varsity level for any of those sports?
One important skill to learn for tennis strokes is to keep your head still during your forward swing. Hopefully this is a habit that you have already developed for baseball (and, perhaps, hockey). Federer and Nadal provide great examples of this. Once they have started the forward swing phase of their groundstrokes, their eyes are glued to the expected contact point. They keep their gaze there with the their head kept very still until their follow-thru is nearly completed. Even on volleys and serves, the head and eyes are kept fairly quiet during the forward or upward phase.
Badminton and tennis have a lot of similarities as well as a lot of differences. When switching from badminton to tennis, it is important to learn what these are. It may take your body/muscle memory a little while to easily develop this. Once you do, you should find that skills from these 2 sports will help you with the other rather than detract or confuse.
A couple of the grips for these 2 sports are somewhat similar, but quite a few are different. For both, your grip should be fairly relaxed most of the time -- many novice & intermediate players grip the racket a bit too tightly for both sports. For badminton, however, the racket grip is more in the fingers whereas, for tennis, it is more in the palm of the hand.
I have taught a number of intermediate/advanced badminton players to play tennis. Badminton players tend to pick up volleys and the overhead (smash) more easily than groundtrokes. Both sports use the core/torso, shoulder, upper arm, elbow, forearm and wrist in a similar manner on overhead shots. There are some differences, but I'll mention them if you request it. Badminton players can take that overhead motion to learn a basic tennis serve motion. Of course, you will need to learn to develop a good service toss for tennis. Also a bit more coil, shoulder tilt and racket head drop is needed for a tennis serve. Various swing paths and arm/racket variations are needed to develop a variety of spin serves as well.
The footwork for badminton and tennis are quite a bit different as you may have already learned. The kinetic chain for strokes is sometimes different as well. For tennis, nearly all strokes will require a full kinetic chain sequence. Knees are bent to utilize leg power. This is transferred to rotations of the hips, core, and torso and then to the shoulder and various parts of the arm (elbow, forearm, wrist, hand). Badminton, OTOH, utilizes a full kinetic chain only when there is time to do so. Quite often, badminton replies are so quick that the lower body components play a much lesser role -- sometimes badminton replies use primarily the shoulder and arm components (and sometimes the core/torso). Tennis grounstrokes (and other strokes) should almost always employ a full kinetic chain.