The rallies were held as lawmakers are due to debate the deeply divisive issue next week.
Opponents of the bill in the mainly Catholic nation are planning to hold their own demonstration next Saturday.
Italy is the only major Western European country with no civil partnerships or gay marriage.
Last year, the European Court of Human Rights ruled that Italy had violated human rights by failing to offer enough legal protection for same-sex couples.
On Saturday, protesters said gay and lesbian couples should be allowed to have civil unions and legally recognised families.
They also said their parental rights should also be enshrined in law.
"They are here. They exist, they are part of our families, and it is unfair that they do not have the same rights as other Italian children," Mirco Pierro, who together with his partner and two children rallied in Rome, was quoted as saying by the Associated Press news agency.
He said he had married his partner in the US, but their union was not recognised in Italy and only one of them had parental rights.
Prime Minister Matteo Renzi has long promised to pass a law on civil unions, and lawmakers are expected to debate the bill on Thursday.
A small number of municipalities in Italy allow local civil unions, but there is a national ban on same-sex marriage and the benefits of the existing provision are limited.
Tens of thousands of people have marched in cities across Italy, demanding legal recognition for same-sex couples.