There are plenty of things that you can do in Goa in 5-7 days. I'd suggest if you can bike it up, then you can also think of visiting Gokarna, 150 kms from Goa. It's an amazing place too. But anyways, I'd like to give you an idea of the things you should do in Goa.
If you are going by train, and you can bike, I'd suggest you get off at Madgaon. From there get towards Madgaon Bus stand. Usually you'll get Bike cabs there for about 50-100 bucks who can drop you to bus stand. Near the bus stand there's this guy called Mr. Pereira, he's pretty reasonable, Bike vendor. Hire a motorbike for around 250-300 bucks, negotiate as it's off season now.
From then on, I'd suggest you directly head to Anjuna. ( There's no point in going to North Goa ( Calangute, Bagha) unless you want to do watersports ( not that great) or hit the pub.
Anjuna has many shacks to stay. You can stay at a beach shack or a shack behind the beach. I went there in July, and stayed at Curlies guest house. Check with them, they also have a FB Page. Anjuna is a good place to relax for one/two days. While you're there, you can also visit Old Goa, visit the Cathredal, the church museum and go to Ft. Aquada. All must visit if its the first time in Goa.
Then you can head towards South Goa, one of the best beaches in Goa. Palolem beach.Should be around 40 kms or even less from Anjuna. It's clean, beautiful and really amazing. You can get to do Kayaking, go to Cocktails & Dreams ( It's a nice lounge place to hang out) And also visit Agonda Beach, and Agonda Fort. Very deserted and serene at this time of the year.
Must Dos.
Drink Kings Beer
Buy a bottle of Port Wine/Any other wine for cheap. ( U can carry 2 bottles at least)
Cashew nuts.
Feni ( Although it'll stink for a couple of days.)
Do Kayaking if it's ON at Palolem.
Visit Curlies in Anjuna Beach.
Go to Agonda beach near Palolem.and Agonda fort. Agonda beach is the most deserted and beautiful beach at this time. Kinda like a private beach.