Tips derived from wikiHow
What you are hoping to receive in the deal (or your own crib) is a sequence (or "run"). The cards making a run do not have to be the same suit. So if you have a 6 and a 7, try to keep them. If a 5 or an 8 pops up, you have 3 points ... plus any "fifteens" that arise. The beauty arises when you have any duplicates in your run, such as a five-five-six-seven. Then you count the "run" twice and top it off with a pair, making a total of 8 points. This is a double run, and if possible it is almost always advisable to keep it (especially if it contains 15's. If you keep 6-7-7-8 you hand is already worth 12 points and could grow to 24 with an 8 cut).
A 10-pointer will be the most likely to pop up as the up-card, for obvious reasons; so if possible, gear your 4 cards to that, saving fives. For example, if you have two 5s, a Jack, a Queen and two aces, give your opponent's crib the aces. A hand that contains a 5 or two cards that sum to five can never score zero.
Try to never place a five nor a pair in your opponent's crib. On the other hand, don't destroy your own good hand just to give your opponent a lousy crib. (They can sometimes make nothing out of something - and something out of nothing ... no guarantees!)
Experts suggest that throwing an off-suited K-10 to the opponent's crib will on average give up the fewest points. A pair of fives is by far the worst.
During pegging, it is best to start off the round with a 4, a 3, or even a 2 - your opponent cannot reach 15 with any of those.
People also often lead from pairs to enable pegging six if the opponent pairs it. There is often a bluffing component where a player wonders whether to play a pair for the points, of play off for safety.
It is usually best to not start out with a five - it's way too easy to match it with a face card which is, of course, worth 10 points.
When counting the points on your cards, start off with fifteens, then pairs or triplets, then runs, then 4 or 5 of the same suit. It is, of course, impossible to have all these at the same time because you can never have a pair of anything of the same suit!