Classroom hunts are a lot of fun - provide students with a list of things that they need to know or be able to locate around the room; have them work with a partner or on their own to locate everything.
The "Two Truths and a Lie" game is another one to get students to share a little about themselves - students do exactly what you would think; they come up with two true statements about themselves and then one lie; the rest of the students are to determine which of the three is the lie; the student who gets it correct goes next; rise, repeat!
I like this easy memory game as students learn each others names and a little about each other.
Get them sitting in a circle. You start by introducing yourself and say one thing about yourself (e.g. sports or foods you like, where you are from, how many people in your family), the next student must introduce you and repeat your fact and then introduce themslves and say something about them. In theory the last person is repeating the names of everyone in he circle. But it gets hard after about 5-6 people, so you can modify it if the group is large.
We do "Bio Bingo" and "Alpha-Seating" Games on the first day of school. I teach 6th Grade Technology Integration so all of my students attended one of 4 elementary schools in our corporation the year before.
The Bio Bing is a bingo game I made up. The kids all get a sheet. Their are descriptions in each box. They are like "Find a student with more than 5 pets. Find a student who has visited another country. Find a student from a different elementary school than you. Find a student who has attended a school corporation different than ours at some point. etc"
When they find a student that meets the criteria, they have them sign their name in the box. First one to five in a row, yells "Bingo!" Then I take up their sheet, and go over it with the class, talking to the students they had sign their sheet and getting to know some students.
The Alpha-Seating is where I have the students sit in Alphabetical order all by themselves. I dont help at all. They have to talk to each other, sharing their names. I will see some kids stand in a corner, and other run around being the leaders and organizing others. It is a great way to see leaders emerge from day one. Finally, I pass around a sheet, they sign their names and BOOM I have a seating chart.
Then we start going over the class rules!!
I'm not sure what grade or subject your teaching, but my 12 year old used Wordle in her class. They were asked to input words that describe them and the program gave them an interesting patern of colors and text that was unique to each student. They did something fun and the teacher learned a lot about each student.