Hi

Sliema is very handy for local busses which will take you all over the island, for some locations you may have to go in to Valletta to get a bus. The local busses are very cheap, fares are mainly 49c (approx44p) although there are some more expensive routes, if you are going to use the local busses regularly it might be worth considering a 1,3 or 5 day pass.
One day ticket € 3.49 each
Three day ticket € 9.32 each
Five day ticket € 11.65 each

Take a look at the ATP website http://www.atp.com.mt for a lot of information including bus routes and timetables.

You are close to Valletta and you can easily get there on a small ferry or take the bus.
If going to Valletta by bus get off at the main bus terminus at the fountain and enter Valletta through City Gate, there is a major new development being built there at the moment.

I would recommend a visit to the Upper Barrakka Gardens which are only a few minutes from the City Gate, great views and well worth a visit, you can buy a small map in one of the small shops on the left as you enter Valletta or ask for directions.

There are an an amazing variety of attractions to see in Valletta, too many to list and the link at the bottom of the page will help choose, I would recommend going in to St John's Co-Cathedral which is very ornate, there is a charge to visit this particular church.

If you want to visit a market there is a daily market held until midday in Merchant St, if you want to visit a large market then take a trip to the small fishing village of Marsaxlokk on a Sunday where there is a large market held, this market sells everything and has a large fish market area.

Other places that I would suggest include getting a bus to Rabat / Mdina.
In Rabat you can go and visit the Catacombs then take a wander in to the 'Silent City' of Mdina. Mdina is a lovely old city that has many small streets and alleys, well worth taking a leisurely stroll around.
There are many fine places to have a bight to eat, for lunch I can recommend a small wine bar called Don Mesquita which is located in Mesquita Square next to the Mdina Experience, lovely local food and wine.

While there you might want to visit the Mdina Experience http://www.themdinaexperience.com/ The trailer will give you a taster.

Make sure when in Mdina you go to the top of the city and experience the lovely panoramic views of Malta.

Close to Rabat / Mdina is the Ta' Qali Crafts Village, this can be done on the same day as a trip to Mdina, you can see glassware and many types of craft ware being made and sold here, including lace, pottery, silver filigree jewellery, ironwork, woodwork and leatherwork.

When at Ta' Qali a visit to the Malta Aviation Museum is well worth a visit, you might have to ask for direction as it is a bit of a hidden gem http://www.maltaaviationmuseum.com/

There are 365 churches in Malta so a visit to a couple of them is worth doing, St John's Co-Cathedral mentioned above is very ornate and most churches are very lovely inside, you may wish to visit the The Church of the Assumption of Our Lady, commonly known as the Rotunda of Mosta or what most tourists call The Mosta Dome.
This church has one of the largest unsupported domes in the world however it is more famous for the bomb that fell through it during the war, the church was full and fortunately the bomb failed to explode.

If the weather is nice and you want to go to a beach you can get a bus up to Mellieha which has a nice beach, don't get off in the town of Mellieha wait until the bus reaches the beach at the bottom or you will have a long walk down

Another nice day out is to get the bus up to Cirkewwa at the North of the island and take the ferry across to Gozo.
Foot passengers pay € 4.65 return, when you arrive in Gozo you can either take a taxi or get on the very cheap local bus which will take you in to the capital Victoria (Rabat) you can then get other busses to have a tour around the island, a trip to Xlendi for some lunch over looking the pretty bay maybe?
In Victoria you can take a stroll around the Citadel and the Rabat and also view the lovely countryside in Gozo, also worth going in to the Cathedral of St. Mary, inside the Citadel, this church didn't have a dome duilt so an illusion of a dome was painted on the flat ceiling.

I don't know if you have children with you but just in case I will mention Popeye Village http://www.popeyemalta.com/
There is sooo much more I could mention so I will finally tell you about about the Hop On, Hop Off bus tours.

Starting in Sliema you can get hop on / off bus tours that take you around the island, there are several tours available, each tour costs € 15 and you get on / off at whatever attraction you wish. An added bonus of this tour is that you get a free harbour cruise included in the price, the cruise doesn't have to be taken on the same day as long as you keep your ticket you can do it anytime and is well worth doing. http://www.exploremalta.com.mt/TourTypeList.aspx?ttid=1
You don't have to book online to get the cruise it is included in your ticket when buying on board the bus.

If you are looking for some nightlife you are not very far from Paceville (pronounced Patchy Ville] which has plenty of pubs / clubs to keep you happy for a week or two, if you want it slightly less lively St Julian's although only a couple of streets away is much more sedate and mature than Paceville.

Take a look at this website for a whole lot more information

http://www.visitmalta.com/main?l=1

Anything else you want to know just ask
Graham