Most nationalities can enter Singapore without a visa. Refer to the Immigration and Checkpoints Authority for current guidelines, including a list of the 30+ nationalities that are required to obtain a visa in advance. Singapore is one of Southeast Asia's largest aviation hubs, so unless you're coming from Peninsular Malaysia or Batam/Bintan in Indonesia, the easiest way to enter Singapore is by air. Changi Airport is officially the 'best airport in the world' and is split into three main terminals (T1, T2 and T3).
Singapore is also linked by two land crossings to Peninsular Malaysia and Singapore is also the southern terminus of Malaysia's Keretapi Tanah Melayu (Malayan Railway or KTMB) network. So, if travelling from Malaysia, the cheaper option is to take bus or train to Singapore.
Ferries link Singapore with the neighbouring Indonesian province of Riau Islands and the Malaysian state of Johor.
Public transport is the best means of getting around singapore. If you are staying in Singapore for some time or are planning to return to Singapore several times in the future, the EZ-link contactless RFID farecard might be a worthwhile purchase. This can be used on buses, trains and taxis.
What to do
Note this: Singapore is a generally more expensive place, compared to the rest of south east asia. So, come prepared for that. If you are into sight-seeing, the options are plenty in Singapore, given that such a small land mass is packed with all kind of scenaries, like skyscrapers (marina bay and raffles regions), nightlife/clubs (clarke quay/boatquay region), cultural hubs (chinatown/little India regions), parks (labrador park, telog blankah), amusement parks (sentosa), legal gambling dens (MBS), Shopping havens (esplanade/promenade), water sports (east coast region and sentosa), food trips (Bugis) and hell, even a legal red light district (Geylang). So, it totally depends on you (and your budget) on what exactly you want to do.