ive vertified hundreds of students innthe cold water if scandinavia and i can givevyou some advice.
buy a mask and a snorkel right away. maybe even fins. you will want that forbthe pool. make sure a dive shop helps you with the fitting to your face. kess volume of air between the glass and your face is better. before you use it the first time smear the inside of the glass of the mask with regukar toothpaste and let it sit for a minute or two. rinse it and clean out thebtoothpaste with regular water. this removes surface ouls and silica from the manufacturing process and will reduce the fogging of the mask.
for your c card id suggest you do ow only. not combined with advanced. do the theory and pool in singapore a week or two prior to going on holiday and doing open water part. id pick my padi center in lets say phuket etc and email them thatbyou would like to come and finish your c card with ow with their instructor. make sure they disclose all fees and if yoy need to do a pool checkup. make sure you know fees for equipment rental and boat fees if any. a shore dive is good enough for open water imho.
you qant your own mask because many nasty diseases soread via the nose and in this climate its better to have your own stuff,
once youve done the pool dive in singapore you know if this is for you or not.
for equipment id suggest consider a regulator and a weight integrated bcd (vest). dont go for the commando back plate stuff, just a regukar bcd. your regulator should have an acto or two secondary stages on it so you can share air with a buddy if required. get a mainstream brand morevfor serviceability when travelling than anything else. oceanic, mares, scubapro etc. you dont need a deep diving kit nor anything with heat exchanges etc. just a devent basic regulator with an easy breath (no labor to breathe, just open the mouth and air should come). make sure there is an easy purge function. for a beginner id skip the adjustable reg flow regulator, just go
for something basic and mainstream.
for bcd just make sure its comfortable. weight integrated is really nice and worth the money.
from here you can consider things like a wetsuit, shorty if you are warm or full if you are cold by nature. i dive shorty but i have a few extra pounds of insulation naturally

fins, are really your choice but consider the style with booties. i have a ski style finstrap which is really comfortable, split fins go faster but cost more.
gloves, just use regular garden gloves and you will be fine.
finally you domt need a computer, but its nice to have. id suggest you start with a watch and just the regular pressure meter from the regulator first stage. should you really get intonit consider a wrist style computer then. an air integrated computer is a bit of risk since you cant dive if it runs out of batteries. air untegration doesnt really give youbthat much anyway,
i dive with a smallish knifevattach to my bcd airhose. its mostly there in case i get tangked in with some fishing line. its jit required but has been useful to me. get a blunt, flat, tipped knife, much less risk of accidental puncture wounds and it doubles as a crude screw driver.
finally, buy yourself the best divebag you can find, this is what you will use and roam through a lot and you need strong zippers which cand handle salt and sun. consider having a couple of pockets for masks fins etc, wheels or not is up to you. i like backpacks more than wheels myself.
good luck and dont forget to always make bubbles!