well, SIN is a multi-racial, multi-religious island, made up of migrant settlers since the early 19ths century. instead of just doing mindless shopping, which u could do so anywhere else in the world, u might wanna consider visiting the cultural settlements of the major races; chinatown-chinese, geylang serai [malay village]- malay, serangoon road [little india]-indians, arad street/beach road-middle eastern/arab... etc. u should be able to find heritage centers easily accessible, where u can learn & understand more about our humble beginnings.
* u'd probably be able to find interesting merchandizes while visiting the cultural settlements. souvenirs & trinkets that u probably won't find anywhere else in the world.... or at least not as well marketed.
not forgetting, u should also try the local food & flavors, because food is something u probably won't get to bring home with u. be adventurous & give your tastebuds something to do.
city hall is also reach with history, where u'd find the conservations of colonial beginnings+ art museum, where u'd find something unique about our arts compared to the rest of the world.
then of course, u'd wanna visit the new developments of SIN, so u don't really go home empty handed. i.e.: the esplanade, the war memorials, the 'old' supreme court, raffles hotel 'the old lady', suntec city, marina square... venture the whole of orchard rd, starting from the junction of tanglin/ botanic gardens, all the way down to cathay (u'd be bypassing the istana as well)
p.s.: if u happen to lose your way, the very efficient public transport system serves as simple travel dirctory (bus routes provided), u could also take the subway aka MRT. in any case, 90% singaporeans are english speaking, so no worries if u need directions.
happy travelling! cheers!