Mobility is a word which has been in the buzz quite recently. It refers to the ability to have a flexible body which can perform Yoga and Gymnastics like a piece of cake. But many people have equated mobility with circus acts that can be performed to amuse others in the gym.
As long as you can perform the basic lifts and cardio with ease, and perform better in everyday life, mobility has nothing to do with performing splits that make people go aghast. It has nothing to do with the general conception that weight training without any aerobics would make your muscles stiff. Research has shown time and again that strength training exercises, when performed correctly with full contraction and expansion, lead to enhanced mobility and flexibility throughout the body.
So you are advised to start lifting in the gym, as lifting when performed correctly with full ROM (Range of motion) can actually help you to stretch your muscles more with the added benefit of building new muscle mass. Apart from increasing on mass, strength training can also help to lengthen the muscles by increasing the stretch potential of the muscles with the application of force.
You can opt out for exercises that make you lift heavy for increasing muscle, with the added benefit of increased ranges of motion so that there is no compromise on flexibility. A good example of such an exercise would be inclined dumb-bell bench press. This exercise when performed correctly would add mass to the upper chest, while also opening up the chest by stretching the pectoral and deltoid muscles.
Similarly, you can target other muscles in your body and devise a routine for each muscle group in a week. But do keep in mind to go for as much range of motion as possible, and go heavy on the weights.