There are two different aspects to consider with this question: the legal and the social.
Legally speaking, manumission (freeing an enslaved person) was relatively easy for a Roman and could be done in several different ways. In one form of manumission, the owner placed their hand on the enslaved person's head with a witness present, said aloud "I wish this person to be free," then removed their hand; the moment the hand was removed, the enslaved person became free. The same effect could be achieved by testament; the testator stated that they wished to free certain people, and the execution of the will made them free. Some emperors, mostly in the second century CE and later, imposed some restrictions on manumission, such as requiring a that fee be paid to the state for every freed person or limiting the number of enslaved people an owner could free at once, but practically speaking, if a Roman citizen wanted to free an enslaved person under their control, there was not much stopping them from doing so.
There were also rare cases in which a child born with Roman citizenship was abandoned, picked up and raised as a foundling, sold into slavery as a child, and then was later able to prove that they had citizenship. In such a case, Roman law dictated that they be set free.
The real barriers that an enslaved person had to overcome were not legal but social. Although the process of manumission was easy, not all enslaved people were equally likely to benefit from it. The majority of people enslaved by Romans worked at basic labor on farms, in mines, in construction, or other demanding manual jobs. People in such conditions were rarely manumitted. Enslaved people who worked in the home and had personal contact with their owners such as cooks, housekeepers, and personal attendants were somewhat more likely to be freed. Those who had special skills, such as secretaries, accountants, traders, or gladiators, were the most likely to receive manumission. In some cases, slaveholders demanded payment from an enslaved person before freeing them, so enslaved people in a position to make money by their own efforts, such as artisans working outside the home, were more likely to see freedom, but this practice was a matter of custom only; there was no rule that an enslaved person could buy their freedom. It is impossible to estimate what proportion of enslaved people in the Roman world were freed, but it was certainly well below half, and probably a good deal less than that.
In early Rome and under the Republic, any enslaved person freed by a Roman citizen was automatically entitled to citizenship themselves. The emperor Augustus changed this law so that freed people instead became Latins, a status short of citizenship (Latins had legal rights, such as the right to enter into legally binding contracts and the right to contract a valid marriage with a Roman citizen, but no political rights, such as the right to vote in the Roman assemblies). Freed people became clients of their former owners and owed them obligations of deference and service. In rare cases, freed people who failed to show sufficient deference to their patrons could be legally returned to slavery.
Most freed people were of low to moderate social standing and faced some amount of social stigma in Roman society, but there was no absolute bar to freed people or their families rising in social status. There are some known instances of the children of freed slaves mixing with the elite of Roman society--most famously the poet Horace--but such cases are the exception, not the rule.