After launching the game and choosing your name and password, you will be given an option to start the tutorial, and will need to complete the test mission assigned by Uplink. There is no threat for arrest within the test mission, but after the test mission, you will risk this chance even more.
To complete the mission, you will need to purchase the password cracker from Uplink Internal Services, and bounce your connection across multiple hosts. To bounce the connection, click on the map in the top-right corner, and click on each white square before choosing the test server as the end point. When you click connect, you will get additional time to use the cracker on the test server.
Download the file, and reply to the initial e-mail with the test file attached. The next step is to delete traces of you contact with the server. However, you will first need to obtain the Log Deleter software for this, and you will only afford it after a few real missions.
Note that once you start targeting any other systems, you can have your gateway seized and your funds confiscated. This announcement will appear suddenly and without warning if you leave any logs present for an extended period of time.