This question has been buzzing around my head for a while now. I get confused with Let's and Lets. I usually use "Lets" since I believe apostrophes and "s" combined always mean "is" and that Lets would substitute for "Let us". I could be wrong. Someone, clear this up for me.
Let's is the technically correct version. The apostrophe is a placeholder for the skipped letters, not specifically is.
Let's is correct, as Rose said. It's a contraction, and nearly all contractions use an apostrophe to indicate missing letters. However, contractions can be quite irregular. For example: shan't = shall not (not written as sha'n't, contractions (always?) use at most one apostrophe) won't = will not (where the heck did the 'o' come from?) ain't = am not, are not, or is not (exact derivation is unclear - aye not?)
And just in case it has not been mentioned already, let's is a contraction of let us. Not to sound pedantic, but the un-contracted form is unusual in everyday speech except perhaps in the phrase, “Let us pray.” Edit~ Just read through the whole of the OP, and this was already mentioned. Ignore me. I haven't had my coffee yet!
The others have said it. Let's is "let us", so use it when saying, "Let's go do such-and-such". Lets is a word, too... if you use it as in, "He lets us play with his dog."
Don't you just love English? In the sport of tennis, if a ball is served and it hits the top of the net, falling directly into the receiving player's service court, it's called a "let, or a let service". The referee will usually shout "let" when this happens. The losing player complained, "Come on, ref! What's with all these lets? That last one wasn't even close to the line!" Then he slams his raquet against the ground, shattering it into several pieces, and receives a cash bonus from his sponsor for the extra TV coverage he generated in the news! Yeah, yeah...too obscure, I know. LOL
There is also the plural of the unusual noun let, which can mean: 1. A lease (chiefly British) 2. a voided tennis play (must be replayed) 3. (Legal) an impediment or obstacle
You can easily find out which word is needed by stretching it out. If you were going to say "let us go to the playground", then you would say "let's go to the playground". Same with anything else. "They are going to the beach" and "They're going to the beach". Just use the apostrophe to cut out the letters that aren't needed.