As mentioned earlier, you'll want to make sure your new Mac's hard drive is large enough. Ideally, it should be at least 20% larger than the amount of hard drive space occupied on your old Mac.
The easiest means of transferring via Migration Assistant is via a Time Machine backup. Do you have one? Make sure your old Mac has been backed up just before you run Migration Assistant. (Migration Assistant does provide for transfer via WiFi, but I'd not recommend it, as it's pretty slow.)
Here's how to transfer via Migration Assistant: Transfer to a new Mac with Migration Assistant - Apple Support
Migration Assistant is pretty foolproof. The most common issue I see here is that Migration Assistant will have you enter a login password for your new Mac, and many don't realize you can use the same password as your old Mac. They then promptly forget to write down the new password, and are not able to login to their new Mac.
Personally, I wouldn't have the Apple Store do the migration for you...you can do it yourself easily. If you're transferring 300GB, you'll want to allow several hours for the process to complete.
Hope this helps.