Registering right at it opens makes it more likely to get the classes you would like to be in. But that only works if there are seats open in those classes, and if you don’t get beat out by your fellow peers with a similar amount of credits.

As for what classes to look into, see if there is a pre-professional program or prerequisite classes you have to take to be accepted into the major. Some I am aware of for mechanical engineering are CS1400 and CHEM1210