By bus
Buses for Sliema leave the Valletta terminus regularly. Be warned that the Valletta - Sliema/St.Julians routes can get quite crowded in the summer months. Any number beginning with '6' will take you to or from Sliema and St Julian's.
The bus routes are very much concentric to Valletta, such that if you wish to travel from Sliema to another part of the Island in any direction it will likely be necessary first to travel into Valetta then back out again. This is being reviewed with a view to modernization along with other aspects of public transport.
Please note that in Malta bus drivers own their buses and as such the service is run like a cab rank, leading to some excellent service and some less than ideal. The buses tend to be uniformly ancient.