Most people can handle sealing jobs if they have good quality caulk (i.e, not too watery) and learn to handle a caulk gun. Weather stripping is slightly more demanding, but can still be done without resorting to a contractor. To start sealing, locate cracks and holes and check if they need to be cleaned before caulking. Pay particular attention to duct exhausts that open to the outside and any points where pipes or wires pass through walls. Since sealing can be messy work, practice with the caulk gun and get used to the pressure needed to apply a steady bead.
Weather stripping creates a seal around windows and doors while allowing them to continue operating normally (a vast improvement over sealing windows shut). While there are many kinds, spring metal weather stripping is the most durable type available and comes in bronze, aluminum, copper and stainless steel. Look for products with prepunched nail holes. If you suspect your home is wasting a lot of heat, a home energy audit handled by a certified energy rater is the best way to find where your home is leaking. They can also let you know if you need to invest in better wall insulation, which can result in large savings, though it may require a significant upfront investment and the help of a professional.
Note: even if you can't afford a professional energy audit immediately, you can use the U.S. Department of Energy's DIY home energy assessment guide. The site provides recommendations for finding leaks and checking insulation. Also see, Simple Steps' CO2 Smackdown for a step-by-step guide. And don't forget that you can claim the cost of materials for a Federal Energy Tax Credit (unfortunately, the cost of installation cannot be claimed) and state rebates, may offer some relief as well.