Now things advance a level. Your BART train may be sitting there, but the doors will be closed. People are sitting inside--how did they get there?--but you can't get in. Expect this bizarre situation to last for about ten minutes, and just act cool like the locals. Eventually the doors will probably open; that was just to throw you off a bit. You're on BART! Ok. You must get off at Millbrae because that's where Caltrain is.
Surprise! BART doesn't necessarily stop at Millbrae. Sometimes you must go past it to San Bruno, then cross the aisle and catch a different BART back to Millbrae. The conductor is supposed to tell you so, but often is too indifferent to bother; that doesn't matter, though, because BART's PA system is so old and ratty that you wouldn't be able to hear it anyway. What you'd hear is "shbluwarmillblahcalshulb". If by some twist the intercom started working, you might not understand, because the conductors are bored and like to read the script as unintelligibly as they can. So make sure you're an expert in the right American accent. Anyway, now that you're going the other way on BART from San Bruno, you get off at Millbrae.
Now you have to figure out how to get out of the BART station. That's almost the same as how you got in--you insert the ticket and then it pops out and you take it and it lets you through, unless you paid exactly, in which case the ticket doesn't pop out but does let you through, or unless you didn't pay enough, in which case it pops out but doesn't let you through. In that case you have to find and figure out the machine that lets you add enough money to the ticket to get let out of BART. That's not quite the same process as the one by which you bought the ticket in the first place, but the learning curve is similar so you may enjoy it a bit better.
Congratulations, you're out of BART. Your next mission is to figure out where the Caltrain platform is. It's nearby, haha, but don't think it's easy to get to! I've wandered around looking for it for ages. You have to go up the correct escalator and down the correct staircase; there are perhaps a dozen to choose from. And make sure you end up on the correct platform, because otherwise your Caltrain is going to take you to San Francisco, which is not Mountain View at all. It's far from obvious which platform is which, especially since after working your way through the maze I've described so far, you will have zero orientation as to which way is north. You want south. Actually what you want is "San Jose", because that's where the Caltrain you want is going. You don't want San Jose, but San Jose is the way to Mountain View. Fortunately you knew that.
Now buy a Caltrain ticket. That involves figuring out which of several "zones" you want to go to, each of which is a list of names that mostly mean nothing to you but one of which, you'll eventually discover, includes Mountain View. Then do another simple half a dozen steps to pay and get your ticket. Edit: but don't miss the demonic detail kindly supplied by ghshephard below, unless it pleases you to figure out the hard parts yourself.
The machine will spit out two tickets, though. One is your receipt, which you probably don't need, and the other is your ticket, which you certainly do. Don't make the mistake of keeping the wrong one; if you do... that's an exercise to the reader. Now you're ready for the train. Oh, one passed by while you were trying to find your damn platform and buy your damn ticket, but don't worry: another will be along in an hour.
Make sure it stops where you want to get off, though. If it's an express train, it may blaze right through that stop, in which case you'll be backtracking on a train for the second time today! (I have made this mistake.) However, you're going to Mountain View and I believe the express trains all do stop there. Better double-check, though. But there won't be any signs on the platform telling you which trains stop where. You might try to look it up on your phone, but there won't be any internet access, so be aware which arm or leg you may pay for roaming data; also, roaming data is often so painfully slow as to be unusable, and the Caltrain schedule if you do find it will be a pdf you have to download. On the bright side, this gives you something to do during your hour waiting for the next train.
Once it comes, you'll get on, but since you're coming from the airport you probably have luggage, and that means you probably want The Luggage Car, which is usually in the middle of the train and has a rack you can put your suitcase onto. Since you don't know that, be sure to be magically positioned so that that car happens to stop in front of you and you happen to see a little sign that says "Luggage Car" to the side of the doors. If you suck at being magically positioned, lug your luggage through the aisle of one of the other cars and hope that you find a couple of adjacent seats and that someone doesn't come along who needs the seat so you have to take your suitcase off it. Failing that, sit with your suitcase on your lap.
Finally, be aware that the stop you want to pay attention to is not Mountain View, where you're going, but San Antonio, the one right before it. Because once the train leaves San Antonio you'll be wanting to begin maneuvering yourself and your suitcase near an exit. Those doors don't stay open very long! If you wait to hear "Mountain View" before you start the maneuver (Caltrain announcements do tend to be audible and intelligible, mostly), you may find that the thirty seconds or so that you have before the doors close again are insufficient to get you and your suitcase off the train, in which case, welcome to Sunnyvale! Don't worry, though; all you were doing was going to meet PG at Y Combinator to talk about your startup. Sunnyvale's nice too, and you saved money on a cab.
I've grown fond of Caltrain since then but will forever empathize with beginners.
Edit: Fine, I added whitespace, but if you're going through this process for the first time, don't think you'll get any whitespace.