Actually, it is very easy and quick to get around in Singapore. It has a great Metro (MRT) that is very easy to use. The taxis are also very good. The government has strict regulations so I did not worry about the drivers cheating me. I took them several times both to and from the airport, to the Botanical gardens and also to the Singapore zoo at night. Usually the cost was very reasonable. I was a woman alone and never really worried. Being there for a short time, definitely consider using the taxis. I found them very affordable, but I do realize the NZ$ TO SG$ may not be as good as the US$.
I was staying in the Marina area at the Pan Pacific. There are extensive malls and underground shopping areas that are connected to the Metro that allow you to walk in air conditioning. I was able to go from the hotel into a mall that extended to the metro area near Raffles. From there, I could take the MRT to Chinatown or Orchard Road.
I took a look at the Fodor's on-line guide to Singapore. I did a quick look at the hotels and it does look like there are several budget hotels near Chinatown or off Orchard Road. Orchard would probably be the most central. I think in Chinatown you would probably take the Metro to go to the other areas that you might want to see, such as Orchard Road and the Little Indian area. I took the small mini city guide from Rough Guide and found it great. You can check it
on-line .
I only had a weekend plus two working days. I loved the Little India area. It was so different. I loved the perfume of spices, bright colors and sounds. It was sensual experience. The wet market at Zhu Jiao Centre is great. You do need a bit of a strong stomach to be able to view the butcher area and beware of the smells of the Durian sold in the fruit area.
Chinatown is also very interesting. I did not get to Eu Yan Sang Medical Hall because it was Sunday and closed. But I did get to go into Mariamman Hindu Temple. Be prepared to remove your shoes before entering the Temple. Definitely go walking in the area of Telok Ayer Street. Beautiful restored old buildings. You might also want to check out the movie Indochine. It's a French movie about Vietnam but it was filmed in Singapore and some of the shots was in this area along with an area near Raffles Hotel.
I didn't get to the Arab Quarter. Orchard street has tons of shops and malls. If you are into shopping this is the area but it is rather modern. I prefer experiencing the cultures of Little India or Chinatown. Do pick up the walking guides to Chinatown or Little India from the tourist bureau if you can.
For a very unique experience, you may want to go to the night safari at the Singapore Zoo. The zoo is open after dark and you can walk on semi-lighted paths and experience it at night. If you like zoos, I highly recommend it. You can also take a tram but I liked walking. The best was walking into the enclosed area with the bats.
I think staying at any of the central areas, Orchard Road, Chinatown, Marina would be good. I wish I could recommend something inexpensive. I barely remember the Maxwell road area. I think I exited on Maxwell street from the MRT (metro). It was Sunday so the area wasn't very busy. It was outside of the main Chinatown area and I walked from there to Telok Ayer and on to Bridge street.
Good luck,
Marta