There were so many Messianic Jews in the early first century church of God that religious leaders in Jerusalem considered it a riotous sect of Judaism!
5 We (religious leaders) found this man (the apostle Paul) to be a dangerous nuisance (the Greek literally means a plague or pestilence!); he starts riots among Jews all over the world (actually, it was religious zealots who started riots because of Paul!) and is a leader of the party (sect) of the Nazarenes (Acts 24)
Messianic Jews are not all in one monolithic group with the same set of beliefs and practices. In fact, there exists a wide difference of doctrinal understanding among those considered Messianic.
A Jew who believes in Jesus fulfilling God's messianic promise to mankind may keep most of the observances they did in the past and which others of their former religious affiliation still keep. He may keep God's annual Feast days and avoid eating unclean meats (e.g. pork) and seafood (e.g. lobster, shrimp). He may wear prayer shawls with fringes and other related religious clothing. He may continue in some traditional Jewish observances and customs but add in the beliefs and practices of standard Protestant theology (observance of Sunday, not Saturday, as day to worship God, celebrate holidays such as Christmas and Easter, eat pork and shellfish, etc.).
There are also many groups that label themselves Messianic whose membership is composed primarily of those who are not a Jew by birth. They prefer this label as opposed to strictly calling themselves Christians because they have adopted many of the same customs and traditions that purely Jewish-based groups who believe in the Messiah observe.