Usually I consider this a more or less easy question to answer. I alsways go for the dry season. But in Sri Lanka there are two rainy seasons which makes it more difficult.
For people visiting Sri Lanka the first time the South- West will be more interesting. This is why I recommend the dry season from November to March.
Please be aware that nature doesn't exactly follow our calender. I have been in Sri Lanka beginning of November and I was lucky.
One year later I went to Thailand mid of November and there was still quite some rain. Sure Sri Lanka and Thailand do not get the same weather but rainy season is similar in theory.
So if possible, it might be better to go in December to be on the safe side.
There is monsun (rainy storm) from South- West and North- East, so everywhere ;-)
Sri Lanka is just south of India and north of the equator. This makes it very tropical. There is dry and rainy season.
Dry season in the South- West is supposed to be from November to March. This includes the "Cultural Triangle" of Anuradhapura, Polonnaruwa and Kandy.
The dry season of the North- East is from May to October.
Sri Lanka is a tropical, third world county.
There are "tropical", "exotic" diseases. Malaria is rather rare.
You should see a travel doc before going and get necessary vaccination.
I would say it is typical for Asian third world countries that despite the poverty it is rather safe. I have never felt unsafe in Sri Lanka. But of course there is stealing and others. Watch your belongings, don't show big amounts of money and valuables, if not necessary.
Compared to India I would still rate it safer, especially for women if you consider the media coverage on rape issues in India in the past years. But India is close and influencing and I would still follow the general travel advice for women to not go alone at night in areas which you don’t know.
Maybe you have heard of the civil war which took place in Sri Lanka for many years. It is over. And the former war zone, especially in the north, can now be visited!
I was surprised how easy it was to get around on Sri Lanka.
There are trains running between the bigger places. For example Colombo - Anaradhapura or Kandy - Galle. So best to take trains here. They are cheap.
From Anaradhapura to Sigiriya I took a private taxi which was more expensive. There are also busses but you have to change somewhere in between.
So I would say it is a mixture of train, bus and a taxi, depending on your route.
The train ride in the central mountain area is supposed to be very nice.
You can compare the required budget to South East Asia and that's what I consider a low budget.
If you can find cheap flights, it can be a real bargain going to Sri Lanka.
The only place which I found a bit more expensive was Negombo. I think this was the first touristic developed area on Sri Lanka. It is close to Colombo and the international airport. It is more touristic, not so much a backpacker area.
I didn't find anything which had to be booked or arranged in advanced. I did everything spontaneously once I knew where to go next.
Even if you book things in advanced, try to stay flexible (at least in your mind) since weather or whatever might force you to change your plan.
The great thing about Sri Lanka:
There is everything which you might what to do.
There is culture and history: Anaradhapura, Kandy etc!
There are mountains (Sigiriya, Eliya etc)
There are beaches for swimming and sun bathing (Negombo, Tangalle etc)!
There are beaches for surfing (Arugam Bay etc).
In the North and the East there are less tourists but interesting and beautiful places. :-)
in 2 weeks I did:
Colombo – Anuradhapura - Sigiriya - Kandy – Galle – Tangalle – Negombo
I took it easy. Especially after Kandy. I could have done one day extra in the mountains like taking the train to Nuwara Eliya.
Three weeks would be perfect to see some places on the North or the East coast.