In my opinion, Kolkata is one of the 5 most interesting cities in India. The fact that it is the only city in the country to have a tram network would alone put it on my map of Cities I Love. What's not to love in something that marries the joy of exploring a city with the joy of riding a train?! And I will tell you a secret - this is the only tram in the world that will stop wherever you want, whenever you want. Say your tram is turning the lush corner of the 'maidan', and you see an interesting football match that you'd like to participate in, or it is crawling through the impossible traffic on the M.G Road, and you spy rainbow colored bangles that you absolutely have to try (yes, the tram moves slowly enough for you to be able to admire bangles in stores). All you need to do is smile and nod at the tram conductor, who is busy in a heated political discussion with a passenger, ring the tiny bell next to the door, and the tram will stop. No one will be annoyed, or ask you why - it is perfectly acceptable to get off the tram and go where your fancy takes you. And the opposite holds true as well: if your feet are sore, and you find yourself near tram tracks, just hang around. If they see you waiting, they'll be happy to stop. This adorable randomness would explain why the Calcutta Tramways Company (CTC) has made the tram schedules so hard to find. You can find them here, and plan your trip with the assumption that trams start about 15 minutes before or after they're meant to. I wouldn't rely on them for getting anywhere on time, but for a tourist who has leisure, they're a god-send. My advice would be choose any starting point from the schedule, and to walk/take a cab there.
If you don't like the idea of aimless wandering through the city, hopping on and off trams, there are other interesting modes of transport, each one unique to the city. First, the yellow taxis. Most people agree that Kolkata would not be Kolkata without the yellow cabs. Nearly all the taxis are falling apart at the seams, and the cabbies can be very crusty if you rub them the wrong way (one memorable conversation went like this - Me: Will you go to Hindustan Park? He: No. Me, bewildered and upset because he was the 4th guy to turn me down: Why are you parked here then? He, smirking in a way only the Kolkata cabbie can: Because I like watching the traffic). But if you're very polite, and if they're in the mood, they'll go everywhere, and tell very interesting stories on the way.
Second, the Kolkata Metro. The metro line is a fairly fast mode of travel, and is very easy to understand, being a single line. So a train is either going toward the airport (the Dum Dum), or going towards the south of the city, to the Kavi Subhash station. There are 21 stations on the way, and the frescoes on each station wall are unique works of art. I especially like the murals in the Esplanade station. So if you like efficient sight-seeing, and if you're short on time, it would be a good idea to make a list of the places you want to see and eat/drink at, and then use the metro wherever it fits into the itinerary (it's the first metro in the country, and should definitely make it to every Kolkata itinerary) and cabs everywhere else.
And finally, the circular rail. This one is mainly for train enthusiasts like me, but has two added advantages - proximity to some of the must-see places in Kolkata, and fantastic views of the Hooghly river for a large part of the route. A schedule of the circular rail can be seen here, but this again, is a picturesque but less reliable way of moving through the city. My own affair with the circular rail did not begin well. I saw the map and decided to board the circular rail at the romantic sounding "Majerhat" because it looked like it would be close to the river. But when I reached the station, I learned that the monsoons had changed some schedules, and a train was likely to arrive from the station at 3 PM (it was 1 PM then). I'm sorry to have to say that the station was extremely dirty, muddy and depressing. Not relishing the prospect of waiting there for 2 hours for a train that might not arrive, I decided to take a cab to the next station. Things got steadily worse because the road in front of the station was a mud track, did not have any cabs at a 10-minute walk in either direction, and was completely untraceable on my phone maps. After half an hour of walking through ankle deep mud, I reached a road and got into the first cab I saw, and when the cabbie turned around in surprise, (you're supposed to ask - "XYZ place chaloge?", and get into the cab only when they say yes), I'm ashamed that I squeaked - "Just get me out of here!".
But I soon found that the maps were working again, and asked him to take me to what is almost my favourite road in Kolkata - the Sarat Bose Road. After fortifying myself with two fantastic cups of tea at a delightful cafe called The Tea Trove, I decided to tackle the circular rail again. This time, I headed towards the Prinsep Ghat and struck gold. This ghat, very close to the Fort William, should definitely be on your Kolkata itinerary; it has a porch (built in the year 1843, dedicated to one James Prinsep, who is known for deciphering a lot of historical inscriptions on coins and rock/metal edicts, including the Ashoka inscriptions), set in a beautiful garden from which you can some great views of the Hooghly and the Howrah. Princep Ghat to Babughat is a lovely 2 km walk next to the river, and has some interesting historic relics including the Man O War jetty, and a 40-year old ice-cream shop which the locals seem to love. The train station in one corner of the park perfectly rounds up the storybook feel of this place. So I would think of the circular rail as a thing to see and experience rather than a mode of transport. I would recommend taking cab to the Prinsep Ghat (or the tram that goes around Fort William), and walk around the park until you hear the train coming. Then board the train, ride it up and down till the Bada Bazaar station - this should take about an hour. The views from the train will be worth the trouble.