Hi there
one the issue of entry by sea from Singapore, that will only get you as far as Batam, which does not have anything for tourists unless you are weekending from Singapore. From there you will need to take a flight from Batam to anywhere else you were thinking of going, or boat back to Singapore Changi and fly from there. Its really just a whole lot more convenient to do it from Singapore, and the airlines that fly from Singapore area whole lot more safe than some of the LCC from Batam. Oh and probably cheaper too.
There is a ship that sails from Medan to Batam to Jakarta, it takes about 2 hours from Batam to Jakarta, but its not recommended unless you have something against flying. Again probably cheaper to fly. The LCC's out of Singapore can get you to Jakarta, Surabaya, Jogjakarta, Medan, Padang etc for around US$100 each as a rule, sometimes a whole lot cheaper.
On the issue of safety, well you will be very safe almost everywhere in Indonesia. Hanging out at bars at 2am is not a good idea anywhere, but again probably a lot safer here than in any western country. The roads are not that safe, traffic accidents are very common so if you have a driver make him drive slowly. The trains are slow, expensive and have a habit of falling off the tracks so if you want to get to where you are going on time, take a car or a plane.
If you do get off the beaten track you will have the privilege of meeting some of the most wonderful people on earth, everyone will want to practice their english on you and you will pick up Bahasa Indonesia very quickly. Get the usual language books and phrase books now and start practising, its a lot of fun. Remember that the vowels use the germanic tones not the english ones.
Always carry plenty of cash, credit cards are not easy to use and they convert the IDR to USD and then back to your local currency so you will get hit at least two times on your exchange rates. Anyway its not an expensive place. Tipping is not expected, although a driver will always look for more. You can also get almost anything you want done for about $5.00 so if you need a shoe repaired, or a parcel delivered or something else done by someone else, IDR10,000 or IDR20,000 is going to be plenty for the service plus the cost of the materials.
Food is almost always good in Indonesia. Simple tip is eat where there are the most people, even if you have to queue, although as a westerner the locals will happily give up their seats to you as they will want to make a good impression. You will find Indonesian food not as spicy as many places, indeed the spices are taste enhancers not taste blankets.
Make sure the water you buy is sealed. There are two or three main brands of bottled water, so always look for them.
If you are going to Sumatra make sure you go to Lake Toba, you can get there from Padang or Medan, although Medan is easier. Its just the most wonderful peaceful place you will ever go. forget Bali for peace, Danau Toba as the locals call it is the place to be.
Rainy season is a bit of a laugh really, its always rainy season. But yes it can be more rainy in late feb early march, after Chinese New year, but by April should be fine and getting hotter again.
go with an open mind and a big smile, you will have the best time of your life