Meilahti campus is within the city and thus getting there is easy. Here you will find the best options for doing so.
There is one direct tram line connecting the city centre with Meilahti: tram number 4. Have a look at the timetable and other options at HSL journey planner.
There are a few buses connecting the city centre from various stops with Meilahti.
Here is a list of just a few possible lines to get to Meilahti campus: lines number 14, 18, 39, 218, 280 and 421. For a comprehensive list and timetable check out HSL journey planner, Please enter your point of start and Meilahden sairaala as arrival point.
Once you have settled in and do not want to invest in a bicycle you have the option of using city bike (kaupunkipyörä). For 30 euro / season (May - October) you can use the vast network of bikes and stations in Helsinki and large parts of Espoo. Read more about city bikes from HSL's City Bike website.
What's best you can combine the public transport with the city bike, HSL journey planner will show you results with all available means of transport - bikes included!