Since it is a small city Manama doesn't offer public transport.
You can go on foot around it, but it is not recommendable since the city is not designed for pedestrians. Also, it is very warm in Manama so it is easy to get dehydrated.
A good way to get around is by taxi. However, being the only mean of transport taxis are a little bit expensive. What is more it is easy to get ripped off since most taxi drivers do not use the taxi meter and at the end they ask you for more money than you should have paid. That is why once you get in a taxi the first thing you must do is ask the driver to turn on the meter. Soe reliable companies that can be trusted are Radio Meter Taxis and Bahrain Limo.
The best way to get around Manama is to rent a car. There are a large number of rental companies which offer different prices. You can also even try to bargain for the price. This is the best way because it is more economical. You might need to get used to driving in Manama but in time it will become pretty easy.