Yes, there is a waterway between Tampere and Hämeenlinna. At summer it can be the most enjoyable way to travel, while the views at the lake Vanajavesi are at their best. The trip can be quite long though. Finnish Silverline  offers cruises, which take 8 hours one way.
You can get around by car, by bus  or by foot. For driving around, the city center is quite an irritating experience, with its tight intersections and numerous one-way streets. The services in the city center are withing a few minutes walk, so a car is hardly necessary there. All buses stop at the market square while heading west and at the bus station (at Palokunnankatu near market square) while heading east.
Hämeenlinna has a good system of public transport. You can ride from one part of the city to another with a single ticket of 3.30€, kids (under 12 y.o.) 1.70€ (as of Jan 1st 2013). Kauppatori is the center of Hämeenlinna's public transport system, but be aware that many bus lines go in both directions from Kauppatori. You can use interactive route planner  to find bus routes.
Häme castle is the main attraction of Hämeenlinna. Medieval red brick castle, which now serves as a museum and a venue for some events. Guided tours are available. There are also couple of other museums (e.g. prison museum) next to the castle. Between the castle and city centre there is a large park area Linnanpuisto, where the annual Wanaja Festival is held at summer.
Aulanko Originally a natural park with great sights, equipped in late 18's and early 19's with an observation tower, other buildings of stone, artificial ponds and rare plant species. Today the area offers also plenty of services, including accommodation in the famous hotel and various outdoor activities.
Tank museum in Parola [] Situated some kilometres north from the city center. For those who are interested in military equipment.
Holy Cross Church in Hattula, a little north of Hämeenlinna. Medieval, gothic church, built mostly of red bricks. Inside there are many paintings and sculptures.
Ahvenisto ridge Beautiful ridge area in western side of the city. Beach of the lake Ahvenisto, "Appara" is a popular place for hanging out, especially in the summer. The swimming arena, in which some swimming competitions of 1952 olympics took part, isn't in use though. In the ridge area there are also e.g. paths for jogging or skiing and a racing track which is a nuisance among locals and a synonym for Ahvenisto for others, while many motor sport events are held there.
The birthplace of Jean Sibelius is now a museum, telling about the composer's childhood, at Hallituskatu 11. The old wooden house is conserved between bigger modern buildings. The museum is open after renovation.
There aren't many regional specialities available. If one doesn't count hotsi, a local version of hot dog, which is sold in grill kiosks.
For cheap meal (10€ or less for one person, even as low as 5€ in the lunchtime) you can always find some fast food. In city centre you can hardly miss a pizza/kebab place. There are also Subway, McDonalds, and Hesburger outlets. Golden Rax pizzabuffet offers it's megabuffet for 7,99€.
Popino is located at Keskustalo, a grim building with a set of stairs, which connect Raatihuoneenkatu (opposite of the church) to Palokunnankatu. Popino serves large, quite special pizzas, but also tasty dishes with e.g. steaks and fish. Salad and bread are included in price, so a pizza or "dish of the day" with water as drink might cost you 10€ or so, but also finer (and more expensive) meals are available. Popino has something for everyone. Raatihuoneenkatu 11, tel. (03) 6532555
Piparkakkutalo "The gingerbread house" is easily recognisable of it's white "scaled" wall. With a beautiful view to the church this is one of the finer places in Hämeenlinna. Kirkkorinne 2 (next to the church) tel. +358 3 648 040
If you'd fancy a steak in Finland's oldest pub (or so it claims), check William's pub. The salad bar, included in the price of a steak dish, is quite good. The basic portions with a steak, potatoes and sauce aren't among the biggest or finest available. Rauhankatu 11, tel. (03)656040
From one corner of the market square begins the two-block long promenade street (the first block called "Reska" by locals, the another was built quite recently). There you will find Tawastia Bank and Huviretki (a chain restaurant, which operates next to hotel Cumulus).
At the corner of Sibeliuksenkatu and Palokunnankatu there is Rosso, a restaurant of a Finnish chain, probably with some Italian influence, selling pizzas and other dishes with reasonable prices. Further on Sibeliuksenkatu there are Steak House, Pancho Villa (mexican), and finally at the corner of Sibeliukskatu and Hallituskatu, Coyote bar & grill, another chain restaurant with nice looking menu but not exactly cheap prices.
For some ethnic food, other than kebab, you can go to the first blocks of Raatihuoneenkatu (from market square to the church's direction and on) and find couple of Chinese restaurants. Georgios serves food from Greece and is also located near market square. Linnankatu 3, tel. (03) 682 8884
If you want to have a unique coffee experience, you can go to the Parola panzerbrigade barracks, where they have a cafe accessible outside from the barracks area. The cafe is maintained by the "Sotilaskotiyhdistys" (literally translates as Military Sisters Association) which is similar to NAAFI in the UK, but volunteer-driven. They have by far the cheapest prices around, since the cafe is primarily for the conscripts fulfilling their obligatory service in the Finnish Army. Always pretty busy, with families coming to visit people doing their military service, or with the conscripts passing their free time.
You might not want to go bar-hopping in the suburbs, though it's not necessary, while most of pubs, rock-bars, nightclubs etc. are located around market square area.
Kylpylähotelli Rantasipi Aulanko Old fine hotel with recently built spa. In the middle of the Aulanko area, some 5 km away from the city center. Bus lines 2 and 13 leave from here to center once in an hour. Aulanko area offers also e.g. accommodation in cottages. 
Hotel Cumulus Hämeenlinna Located in the city centre.
Hotelli Emilia Small hotel right in the centre at the Reska promenade street.
Puuhamaa is an amusement park (or maybe more a giant playground) in Tervakoski, about 20 km south of Hämeenlinna. Much of fun for kids, though everyone must pay the 17€ entrance fee, which grants you all the activities: waterpark with steam sauna, boats, racetracks with small cars, and various places for jumping, sliding, climbing and playing. Having a picnic is also possible.