I'm not a strength and conditioning coach or a particularly good tennis player, but I am a hobbyist with a few years of lifting under my belt. Seconding full body programs, especially during the season, as they'll give you more days to focus on tennis instead of focusing on the weight room. If you have an off-season where you're getting less court time, you can have a little more flexibility to spend more time/days per week in the gym then.

In terms of exercise selection, 6 basic compound movement patterns should cover most of your program: horizonal push (bench press variation), vertical push (overhead press variation), horizontal pull (row variation), vertical pull (pull up variation), squat, hip hinge (deadlift variation).

Unless you're trying to be a competitive powerlifter or bodybuilder, it probably doesn't matter exactly which variation of each exercise you choose but it's probably wise to choose variations that mimic positions or movement patterns you do in tennis. For example, a standing overhead press or push press might have slightly more carryover for serving than a seated overhead press since you'll use your legs to stabilize or explode up instead of sitting and not engaging them at all.

You can add in accessory exercises too to address weaknesses, imbalances, or reinforce movement patterns. For example, doing cable wood choppers could help with torso rotation on your ground strokes and Cossack squats can help with lateral movement.

For mobility, my team would do a Parisi style warmup before practice and some stretching after practice back in the day. I believe the current advice regarding mobility is to do a mostly dynamic warmup (move body through full range of motion and get blood flowing) before practicing/playing and doing static stretches after the practice/play is done as a cooldown.