I have had a few experiences that I can share here. First of all, realize that I do not count. I can't see well enough to count with other players at the table. So the most I do is observe, in a very general sense, the preceived balance between high and low cards. I do bet some progressions, however, so I am no stranger to the procedure casinos go through with big bet spreads.
I was backed down once.
It was at the El Cortez in Las Vegas. I was playing a single-deck game, and was betting a 5 - 10 - 25 progression. I'd have bet higher, but I never lost three in a row. It was a slow time, and I got a pit boss watching me almost immediately when I sat down. She was standing off to the side, but doing nothing but watching my table.
After a couple of decks, I was up a decent amount, and I saw her talking with a big goon of an older man in a suit. He came and stood behind the dealer, who was a short foreign lady. He watched me lose at $5. Then he watched me lose at $10. Then when I slid out the green chip, he stepped around the dealer and pushed it back to me.
He said something like "I've already backed you down once today. I am not going to have you counting cards here." I tried to explain myself. What I intended to say was "Sir, there must be some mistake. This is my first time in this casino." (which was the absolute truth.) However, as soon as I said "Sir," he cut me off and said "Don't you sir me." I knew right then that I wasn't dealing with the brightest bulb on the tree, so I got my chips together and cashed out up $50 on about fifteen minutes of play.
I have since learned that my $300 buy-in was probably the thing that got them interested in the first place. So the fiorst lesson for me was, when playing at a real low-roller joint, buy in for small amounts. I didn't think much about the woman watching me, because I am fairly accustomed to that. However, I did know that something was up when she went to talk to monkey man and he came to the table. In hindsight, I probably should have got up then before losing the $15 to them.
I had a pit boss count down the discards once.
At a different casino, playing a single-deck game, I had a pit boss come to the table and count down the discards after I won a big bet. I assumed that he wanted to see if I knew what I was doing or not. When he was finished, he just put them back and walked away. I didn't see much of him for the next two hours, which were a nice win for me. I assume the count must have been against me when he checked it. While he was there, I just ignored him.
I've been asked how much I won a few times.
Once in a downtown casino, the pit boss came over to me after a nice long winning session at the nickle double-deck game and ask how much I had won. I told him $350, and chatted him up a bit about the town. He seemed very nice; and I have no reason to think he suspected me of counting. He had, however, watched my play quite a bit during the session.
Another time, a pit boss asked me how many black chips I had taken away from one table when I had moved to another one. He then insisted that I count them in front of him instead of him counting them, even though I pushed the stack over to him to count. I have felt since then that he was doing that to try to screw with me, in case I was counting. And interestingly enough, I went on a losing streak right after that and dropped most of those blacks right back to them.
I had a dealer overly-interested in my strategy once
I was playing, and winning, and a relief dealer came in who just made it his business to discuss my strategy at great lengths with me, even though the table was full. It wasn't much later when I got yet another new dealer and proceeded to lose a good bit of what I had won previously. I have since learned to be more evasive. It is one thing to make a comment or ask a question here and there. It is quite another to try to learn all of the details about what I do in certain situations.
In general
You should expect the dealer to notify the pit when you raise your bets beyond a certain level, which varies from place to place. They normally do this by calling out "Cheques Play" and then getting some sort of response from the pit. I played one day when the dealer called that on practically every single hand for four hours; and I hardly ever saw a boss. Other casinos don't call it at all unless you have a huge spread. Once I was at such a place, I raised my bet slowly over several hands from $10 to $500. The dealer called "cheques play" on the $500 hand, and the pit boss came over to watch that hand.
Many places will also call out if you change the color of your chips. For instance, if you start betting green chips at a $5 table where you've been betting red, they'll sometimes call out "green action" to the pit. And most places call out "black action" when you bet $100 chips. But this isn't "heat", necessarially. It is just them notifying their boss according to their protocol, and it sometimes has more to do with accounting than counter-catching.
Real heat starts when they start shuffling up when you put out a big bet, changing dealers on you when it isn't the appropriate time to do so, etc. You have to just play on when you see someone watching you. If you act all freaked out by it, they may take you for a counter because of that. You have to remember that the average weekend gambler does not pay attention to the pit. So if you do, you're not an average weekend gambler in their mind. Plus, sometimes when they are watching you they are trying to decide how much of a comp to offer you.
In my personal experience, the "heat" is greater in Las Vegas than anywhere else I have played; and it is much greater in single-deck games than in any shoe game anywhere. I assume that is because they are more vulnerable there, and that most counters seek out those games.
Lots of books have some info on casino heat and countermeasures. I always recommend Blackbelt in Blackjack as a good starting point.