Hello Lavi.
Babies in Sri Lanka go through a series of vaccinations. Quite a few pediatricians around too. I can speak to my sister and get the name of the pediatrician she consults when her kids fall ill. These pediatricians will also be able to give guidance on the vaccination schedule you should follow.
As you rightly said, dengue is still a big problem. You need to make sure you keep those pesky mosquitoes out. I have found citronella oil generally works. Also effective are the local camphorated balms. The strong smell keeps mossies away. There are other precautions you could take - essentially notnallowing mosquitoes to breed in and around your residence.
Baby formula of different types are available. If you have a preferred brand maybe you can let us know and someone can check its availability.
You hould be able to find an English speaking nanny. You will have to advertise locally, especially in the Sunday papers. Not sure what the pay scales are. I have to warn you though - that English is not the first language for a vast majority of Sri Lankans. Hence, the standard of English that your nanny would speak may not be upto the mark. However, I am sure she will improve fast, once she starts conversing with you.
Furniture and nurseries are not too expensive unless you have special requirements and are looking for international standards.
The postal service in Sri Lanka is fairly efficient. Internal mail generally gets delivered in a day. 2days if the addressee is in a remote area. Foreign airmail gets on a plane asap. I have used it extensively and have not had any bad experience with the Sri Lankan postal service to date.
Most telecommunication companies have internet packages at different speeds and cost structures. My folks use an unlimited internet package from Sri Lanka Telecom (the main telecom provider) and the monthly cost is about USD25. Its not lightening fast, but is fairly fast and serves a number of devices at home. Skype calls and streaming video work well, so the connection must be good. 
Any other questions please ask.
I wish you, your hubby and your (expected) baby all the best and wish you all a pleasant stay.