short answer: go to raffles hospital/medical and they will sort you out. (i do not work for raffles and am not affiliated to them)
long answers:
GPs should (theoretically and practically) treat any human patient. you just need to let the counter staff know you are there to simply get a referral letter for your baby - that he/she is not sick and does not need immediate medical attention.
1. Do GPs take care of infant vaccines at all? Or do we have to get referred to a pediatrician to get my newborn vaccinated?
some GPs might, but private clinics generally don't maintain all vaccines you might need at the time you need them. you would be well served by locating a paed that you can go to regularly.
2. If I get a referral, am I likely to be able to make an appointment with a pediatrician in 3 weeks? Or is the waiting time much longer in general?
3 weeks - maybe. depends on the doctor's popularity. walk-ins are also possible if there is an actual specialist in the clinic. your best bet is to find a paed located "in the heartlands" (as opposed to a specialist medical centre) they are generally more accommodating. their address is usually "Blk 123 #01-3456" or somesuch.
3. Would it work if I just walk into one of the clinics/hospitals to request vaccination for my newborn? If so, what clinics/hospitals can I go to? And roughly how expensive is it without insurance coverage?
look at my first sentence and reply to #2.
i quote
http://www.hpb.gov.sg/HOPPortal/health-article/2920
What Are "5-in-1" and "6-in-1" Vaccines?
These newer childhood vaccine formulations combine vaccines against 5 or 6 diseases into a single injection.
The 5-in-1 injection combines vaccines against Diphtheria/Pertussis/Tetanus (DPT), Polio and Haemophilus Influenzae type B (Hib) in 1 injection.
The 6-in-1 injection combines vaccines against Diphtheria/Pertussis/Tetanus (DPT), Polio, Haemophilus Influenzae type B (Hib) and Hepatitis B in 1 injection.
if i remember correctly, your cost will be well under $400 for either (split over a few visits). others are at an extra cost.
PCV (pneumococcal) is ~$150. hep A ~$70, hep B ~$30 (same price as twinrix at ~$100). rotavirus ~$90, infuenza (at the right age) ~$25, chicken pox ~$75. MMR ~$130. consultation charges are extra.
4. Will I be able to request for a specific pediatrician to be referred to if I find good online reviews? Or is it random assignment?
don't trust online reviews. be warned that referrals are based on loose affiliation/fees, and does not mean much.
find a paed you feel comfortable with. read up on whether ALL vaccines listed above are necessary...
i am not a parent. so i can't answer much else. good luck!
edit: i found this GP's website that lists prices for some of the above. i do not work for them and am not affiliated to them.
http://www.shimclinic.com/singapore/vaccine