Most of the major sights of Kuching are within easy walking distance of each other, so picking a hotel in an area nearby such as the waterfront will allow you to walk to all of the major attractions. Many of the hotels do provide shuttle services to and from the major tourist sights, so check with your hotel about their services.
If you'd like to get further afield, with less effort, then you might want to rent a bicycle. Borneo Bicycle Hire has good quality bikes for hire. Weekday rates are 33 Ringgit (6.72 USD) with higher rates for weekends and holidays. All bikes come with emergency repair kits, rain gear and maps. Helmets are also for rent, and full insurance is also available. Borneo Bicycle Hire is located in Tun Jugah Plaza, near the Hilton Hotel. They are open every day from 9:00 am to 7:00 pm. The telephone number is 019 4844 393. They don't appear to have an official web site.
You'll probably need taxis just to get to and from the airport. There's a flat rate for taxis from the airport of 26 Ringgit (5.29 USD) to any in town hotel. To get a taxi, you purchase a coupon from the counter in the arrivals hall before proceeding outside to the taxi queue. For more options, see the airport transfers page. Taxis do have meters but they don't seem to like to use them, and have a reputation for over-charging customers, so be sure to negotiate the fare before you get in.
While most of the sights are on the south side of the river, there are a few, such as Fort Margherita, on the north shore of the river. To get across the river you can use the traditional river taxis called tambang. The fare for a river crossing is 50 sen (0.10 USD) during the day or 1 Ringgit (0.20 USD) at night.