Fun things to do in North Central Province

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As Eric said, Hotels in Colombo aren't cheap anymore! There's now a minimum established rate for the Colombo (4,5 star) hotels and it ranges from $85/- Upwards(For a good hotel). There are many low-budget decent hotels along the "marine drive" (3km from the Colombo Fort) that would certainly interest you!

As Viajes said, do remember to cover the cultural triangle (Sigiriya, Anuradhapura, Mihinthale, Polonnaruwa, Dambulla)... Also add the Wasgamuwa National park onto your list.. Allow 2 days for the above and head towards the Hill country for a few relaxing days before heading to the coastal regions.. You can check the following tour operator for assistance (www.jetwingtravels.com). I personally love their food as they manage the meals according to Sri Lankan cuisine!

If your in the Hills, remember to drop by the Tea Factory (in NuwaraEliya)and The Haggala Botanical Gardens as its in full bloom during the summer!... The hill country is generally considered lazy as there's no ready made adventures available! - However I personally like to make my own (Dip in the freezing waterfalls; hike up hills; photograph mousedear, leopards,wildboar, sambar, birds, giant squirrels; trek through the vast plains of Horton; Dive in the lakes; Visit the milk factories; cuddle baby goats; road trips; train trip; walk through jungles, tea estates, farms, river banks; Drive through winding roads, shiver in the rain) :)

If you hire a private van (With A/C), the cost will be about Rs.40/- per Km. The cultural triangle will be 800km (round trip from Colombo)..

However If your friends are active, My suggestion is to:
1) Go by train to Anuradhapura ($8 per person)
2) Check in to a low-budget Hotel (with pool).
3) Hire a Van and visit the Cultural Triangle.
4) Get to the Hill Country by Van(Kandy) - The hill country is a vast network of rolling hills and getting from point A to B will consume time. This too can be done by Train, but requires a change at Polgahawela (Which would eat into your tight schedule)

5) Once all activity is completed, head back to the coastal regions (south or south-west). Mirissa or Unawatuna(Galle) which are less populated, less polluted, more private place to hang out.
6) Eventually head back to Colombo where you can put all things together.

If your into Wildlife, Exotic, Romantic or Cultural tours (Do check the above JetWings guys) as they have yeeeaaars of experience! (Plus they work next door).. I cannot vouch for the price, but I will vouch for the quality!

As for your question: Yes! You can go on your own without a Tour Operator, however you will get lost in the commotion and probably cheated into paying something you really didn't need!(There's paper work involved).. If you opt for this choice, there are many things you need to know and avoid. The culture in that area is completely different, the people are different, and the place is Hot and Humid. The Tour Guide will generally know "what to say', "what to do", "where to go", "what to avoid", "where to stay", "what to photograph" etc etc..

What you could do is, customize the tour to include the cultural triangle only. Then do everything else on your own!!
Best of luck!