You don't want to be that guy with the limp handshake. Train your grip strength and not only will you shine on first encounters - as an added bonus you'll be able to lift more! Let us look at ways to increase your actual grip strength in addition to my piece on grip accessories.
Fat Grip Training
Training with an exceptionally large bar forces your hand to grip even tighter than usual. It is an incredibly effective way to build up your forearm size and improve your grip. Unfortunately many gyms do not have wider barbells but you can compensate by attaching a pair of fat gripz. Another useful alternative is to wrap a towel around a pullup bar which increases the bar diameter and forces you to work that much harder.
These are quite common online and in sports shops. They are usually a metallic springy contraption that you can grip and close with one hand. They are pretty self-explanatory, but don't forget to turn the grips upside down to work the full range of your grip muscles.
There are loads of exercises you can do to up your crushing power. Try Grip Hangs where you hang a towel over a pullup bar, grasp either side and hold on tight! Farmer's Walks are very useful. Just pick up some very heavy kettlebells or dumbbells in either hand and walk up and down the gym. Even better, load up the hex bar and do the same. Essentially any exercise that involves grip will enhance your grip! I know - rocket science eh?
Hold your arms out in front of you, and pinch two light barbell plates between your fingers. See how long you can last. It is not as easy as it sounds.
Remember that lifting heavier weights will naturally improve your grip strength as you grapple with those 20kg plates. Lift Heavy - Improve Your Grip - Lift Heavier. The circle of life.
But don't limit your lifts to just your grip. Using weight lifting straps will help you lift greater weights. Once you take the weight lifting straps off you will find yourself even stronger than before.