I've tried to find some information about it, but failed. Let's say there is an educated slave living in a relatively rich household of Rome. What would prevent him from fleeing, going to some smaller town of some province to live there as a free man? It's not like they had lists of all free citizens, neither did they have proper identification documents, such as a modern-day passport, which would identify them. And not every single slave wore a collar (which still can be broken).
Well, I suppose it's a matter of means plus motivation. If you're educated - read/speak Greek and Latin etc - then you'd be valuable, and only the psychopathic master would mistreat a valuable peice of property. And you'd need money to get away - some slaves were relatively wealthy, but stealing from your master would be dangerous, the penalties could be horrendous. Then, if you got to this new town, how would you live? If you were educated, even if a slave, you'd consider manual work beneath you - and, of course, slavery has the effect of reducing the value of free labour, why pay someone when you can own them? He (it would be far more difficult for a woman) could become a schoolmaster, but that was no sinecure in the Roman world, living in a cramped tenement, always looking over your shoulder in case you were discovered. You're right, very few slaves actually wore collars, although badges have been found with inscriptions like "1belong to M Julius Horrendus, if you find me, send me back." - which seems a bit daft, as presumably, if they were just badges, they could be removed! And - I'm no expert, but suspect there was a cultural aspect here - if you were a slave, then it was because the gods had made you one.
Interestingly, the Senate grappled with this problem, there was a move - sorry, can't give a date - to decree that all slaves had to wear distinctive dress. It was abandoned out of fear that, if the slaves realised how numerous they were, they'd rise in rebellion.
So, for the educated slave, without a psychopath for a master, engaged in teaching the children or dealing with his master's business affairs, it was probably better to wait and hope he would be able to buy or be given his freedom, rather than risk being caught, flogged or worse, and sent to the salt mines!
As I said before, there could be genuine affection between masters and slaves - much the way 19th century servants in the UK felt about "Miss Alice" or "Master George". During the US civil War, a slave boy was sent to join the Confederate Army with the young master. His master was killed, the slave buried him, wrote a letter to his parents - then high-tailed it to the Unionists to fight for freedom!
If you haven't, you might like to read Plautus' play, "The Prisoners", which gives a very interesting insight into slave/master relationships.
Although, as you say, a rich slave might be able to engineer an escape, most slaves were not rich and not educated.
Slaves could generally be immediately recognized by their dress. Although there were no laws mandating dress for a slave, they tended to wear clothing which set them apart. For example, no slave could wear the toga, so if a man is wearing a toga, you know right off the bat it is a citizen. Also, the tunics worn by freemen tended to be a lot nicer and more expensive than the plain tunics worn by slaves. Likewise for their footwear, which was expensive. Slaves often went barefoot. In general, if you saw a guy with no shoes wearing a plain tunic, it was high probability he was a slave. Freeman also wore hats, called a pileus, which slaves generally were not supposed to do (see below for more about this). Also, there was an ethnic component, since slaves were foreigners, not Latins.
Slaves also had a different, much cruder, speech than the upper classes, which is often parodied in Roman plays. The educated in Rome studied diction and speech extensively in school and spoke in a way completely different than the slaves. If you read Plautus' plays, for example, you can learn how he parodies the speech of slaves. Also, note that in plays the free always wear togas and the slaves always wear tunics so the audience knows which is which.
If a slave was freed, he shaved his head to be completely bald, and then wore a plain (uncolored) pileus, which was a felt cap. Some citizens wore this cap, but always colored or decorated. A man with a bald head and plain pileus was a manumitted slave.
As for runaways, fugitivi, this was always a small problem, but it was hard. In those days travel was expensive so people tended to stay in one place and strangers stood out. Where would you get the money? Usually you needed to have permission to become a citizen of a town, so if you just ran to some random town and applied for citizenship, the authorities would want to know who you were. People abroad on highways were regularly stopped and either had to have a passport or a really good story. Remember the highways were the property of the state in those days and were controlled by the military. Random people were not allowed to just waltz along highways the way they do now. There were severe penalties for people who hid or took in fugitivi as their own, so though it did occur, it was an illegal and risky move. You stood a good chance of getting ratted out by your neighbor or an enemy if you tried hiding a fugitive.
Finally, there were the dreaded fugitivarii, the slave hunters. These guys were professional fugitive locators and they were very good at their job. They had spies everywhere. Not only could they track a particular slave down, but they would grab you if you just looked like a fugitivus. Then they would torture you to reveal your owner. If you gave a fake name, they would just torture you again until you fessed up your real owner. Then they would take you to him and demand a fee from him for returning you.