Erin,
Assuming your train from Paris arrives on time, you should be able to connect with the 06:10 departure from Milano Centrale (but don't stop for coffee). According to the current Trenitalia schedules, the transfer time in Levanto is six minutes. Levanto is a smaller station, so a short connection time shouldn't be a problem.
In this case, I'd purchase tickets in advance for the reason that if your train from Paris is a few minutes late, you won't have a lot of time to buy tickets. There are ticket Kiosks at track level, but if you want to use the staffed ticket office, it's two floors down and there's often a queue. To reach the ticket office, you'll also have to leave the trackside security area, and if there's a queue getting back through the gates, that will add to the delay.
However, there is a slight risk in buying advance tickets. The first train you'll be using from Milano Centrale to Levanto is an Intercity train, which requires compulsory seat reservations which are specific to train, date and departure time (the reservations are provided with the tickets). If your train from Paris is late and you miss that Intercity train, the ticket is worthless and you'll have to buy another (that's assuming the ticket you bought is a Super Economy ticket).
The second train from Levanto to Corniglia is the Cinque Terre Express, a Regionale train. That will come pre-validated and can be used anytime within four hours of the departure time listed on the ticket. If you buy Regionale tickets locally, you MUST validate (time & date stamp) before boarding the train on the day of travel or you'll be subject to hefty fines which will be collected on the spot!
If you decide to buy advance tickets, I would highly recommend www.captaintrain.com as someone else mentioned. They have a very user friendly website, and you can print the tickets at home. They also sell tickets for the French and German railroads (among others).
In the same situation, I'd probably buy the advance tickets, but be prepared for the eventuality of having to buy more if the Paris train is late.
Are you planning to use the Shuttle Van to get up to Corniglia from the station, or walk up the ~400 steps with your luggage?