December and January are the peaks of the rainy season, so there may be some flooding in low areas, like Medan. The rain may not be that bad, maybe just one or two showers a day, so you may not have any problems at all.
If you start in Medan and only have eight days I would suggest you concentrate on North Sumatra. One night should be enough in Medan, if you don't like Medan go straight to Bukit Lawang, you will need at least a couple of days there if you like to visit the Orangutans and do a trek in the national park.
After Bukit Lawang you can go to Berastagi, to see traditional Batak villages, hiking on a volcano and visit hotsprings. One or two nights should be enough here.
That would leave three-four days (minimum) to explore the Lake Toba area. If you are short of time you could always skip Berastagi, Toba is a place you must visit. Lots of cheap accommodation in Tuk Tuk, half an hour by boat from Parapat. Explore Samosir Island and enjoy the lovely climate and scenery around Toba.
I think this should be enough for eight days. Read this article: The North Sumatra Triangle
and this one: Bukit Lawang for more info!
For general info on cost in Indonesia see this article: Money talks, costs and bargaining.
The following is a reply I gave to an earlier post with a similar question regarding cost in Medan (it is valid for the rest of North Sumatra as well):
On Java, Bali, Sumatra, Sulawesi and Nusa Tenggara you can survive on as little as US$10 per day, on a mid-range budget you may use US$30 per day, for US$50 per day you can travel and live quite comfortably. Alcohol is relatively expensive in Indonesia, if you like to have beer to your meal every day you must raise your budget accordingly. A large bottle of local Bintan beer cost 12.000 to 20.000Rp.
If you stay in a budget hotel you can spend as little as 25.000Rp per night for a decent place, go to the area around Mesjid Raya, the big mosque, and Yuki Plaza, there are several good alternatives there, but also more expensive ones. Danau Toba Hotel is another popular alternative, it's not a typical backpacker hotel, and good value for money from about 180.000Rp. You can of course also find more expensive hotels in Medan. If you plan to stay longer, an apartment can cost anything from 6-7 million Rp per year, to US$ 2000 per month.
Transport in Indonesia can be very cheap, a bus to Bukit Lawang cost only 5000Rp. Public and tourist minibuses are more expensive, but still relatively cheap. Even taxis are cheap, but you may have to bargain to get a good price.
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