snow,
The trip from Deiva Marina to Milano Centrale appears to vary between 2H:57M and 4H:10M, so you may want to check the Trenitalia website to determine which train you want to use. The train I'd probably take is a departure at 07:53, arriving Milano Centrale at 10:50 (time 2H:57M, one change at Rapallo).
When you arrive at Milano Centrale, go to the left (with your back to the tracks) and down the stairs to Piazza Luigi di Savoia, where you'll find the Malpensa Shuttle (Bus). You can buy tickets right there and the trip to MXP Terminal 2 will be about 50 minutes (subject to traffic of course). You'll need to validate your ticket prior to boarding the Bus. One other point to mention is that I believe there are two firms offering Shuttle Bus service to MXP. AFAIK it doesn't matter which one you use, however each of them issue their own tickets.
EasyJet now uses a system where passengers check-in and print their boarding pass up to a month prior to departure. There's no check-in at the airport although there is a luggage drop. You might consider paying extra for their "expedited service" as it makes the process much easier. If you think your checked luggage will be over their allowable weight limit, it will be cheaper to pay for the next weight tier at the time of booking, rather than at the gate. As with most budget airlines, passengers are packed in like sardines, but it will get you to your destination.
When you arrive in Barcelona, using the AeroBus is about the cheapest method to get into the city. As I recall, the airport stop is about a block to the right when you exit the Terminal. The website will provide the details (or follow the herd as others from your flight will likely also be using the Bus). The Bus is fairly basic, but as I recall there are large luggage racks in the centre. You'll arrive at Plaça Catalunya, at one end of La Rambla. if your hotel is in that area, you should be able to walk from there.
Happy travels!