You won't become "stiff" from lifting weights unless you gain crazy amounts of mass. So long as you warm up and stretch properly (which should be a law for athletes) you'll have no problems with stiffness.
Funtional strength can be increased in a number of ways; odd object lifting, kettlebells and the like are certainly 1 way to do it but are certainly not a be all and end all.
Athletes who want the best their body can deliver will combine these methods with free weights, bodyweight and even machine weights.
The best free weight lifts to use, as has been said, are compound lifts, such as bench press, deadlift, squats, pull ups and dips etc.
I find my strength increases with weight training, but my power (here I am describing strength as lifting and pushing/ pulling abilities and power as explosive style strength and potential exertion of force through sudden motion) does increase better with odd objects and kettlebells and bodyweight exercises.