Please advise what clothes I should take.

welcome.

first off, SG is the most westernized city in the East. Conservative? that totally depends on someone's POV. I think it appears polished on the surface but in general people are mostly liberal about many things.

It's always hot and humid. not to worry, inside buildings, in public transport etc, aircon is maxed and in some cases, it feels really cold inside. Some places/restaurants do have smart casual dress codes, so if you are planning these hang-outs, at least shirts and pants and shoes would be necessary. pubs, you can go in relax, shorts and flipflops are ok in most cases.

It is a tiny country but is growing larger :-) (reclaimed land). eating, no problem. cheap food courts everywhere at almost any time of the day. It's a city-state, so not much rural areas (if you understand farms and villages) but lots of green spaces around. very easy to go on around, it's a well connected city. bus/ferry/train/budget air to almost every destination in SE Asia.

Thank you ruddy24!
I know what you mean. I experienced every summer in Japan the difference in temperature between the inside and the outside. It's like walking into the air wall. I would shiver as I walked in and choke as I walked out.
In Sept, are you still in t-shirt all day outside?

Yes, there isn't much difference year long in terms of temp usually between 26 and 32 Celsius

The only reasons in Singapore for wearing long sleeves or pants are
- if you are too conservative to show much flesh (most locals aren't)
- at formal occasions (not very common)
- to prevent sunburn or mosquito bites
- or in places with full-blown aircon (which is basically everywhere in-doors)

Thank you! Looks like it's gonna be too easy to pack this timme. A pair of jeans, shorts, a few long sleeves and just enough t-shirts.

How are you going to stay for months? You have a job and employment visa prepared?
(Visa runs do not work in Singapore.)

I might be able to stay at somebody's place but if can't I will stay at backpackers. But it would become very costly to stay at backpackers for long so I would have to pack and leave in that case.

I like to skydive though, there is no drop zones in Singapore due to air congestion so I would need to go to Malaysia. If I go to Malaysia and come back, will I get frowned upon?  I don't mean to do visa runs nor break laws in the country. But I would not mind picking up some casual jobs if available for some bread.

Yes, visa-runs (or anything that looks like one to the authorities) are a no-no in Singapore.
If you want to stay longer than the original visa-on-entry (15, 30 or 90 days, depending on where you come from), you must apply for an extension at ICA. Be prepared to state a very good reason for staying that long, or it will be rejected.
Casual jobs are illegal for foreigners in Singapore and it is definitely not a good idea to break the rules here!

I know I do not want to do anything that gets me into trouble. Law inforcement are not soft there.Is it hard to get a work visa if I mangge to find an employer?

It has become much more difficult in recent years to get a work visa, but is still possible if you find an employer willing to pay you the minimum S$3000/month and if you have a recognized university degree (no chance without degree). This is for full employment only, part-time or temporary work is not possible.