Pay special attention to the air handlers - the metal boxes that contain the blower and other HAVC elements - of your heating/cooling system, when sealing the air ducts.
Many of the worst duct air leaks occur at the air handler and at the joints between it and the main return and supply duct branches. These areas are located close to the blower, and under high air pressure, which increases air leakage through them.
HVAC contractors use special fans,called duct testers, to pressurize the ducts and to measure and identify the leakage areas. Though finding and sealing duct leaks is a relatively simple project, your should consider professional assessment and sealing you aren't motivated or at ease with the physical requirements and hazards of this type of projects. Sealing the air handler and the areas adjacent to it is relatively easy, as long as your know how to access them. Start your work with them.
applying duct mastic
You may apply duct mastic on the seams using a paintbrush, but to seal larger openings it's easier to use rubber or vinyl gloves (consider two or more layers of gloves to protect your fingers from sharp hedges) and to smear a handful of mastic onto the leaky areas.
Joints wider than 1/8 inch or subject to movement should also benefit from reinforced mesh. It's a messy and dirty job. Duct joints should be fastened with screws.
To seal joints between flex duct and plenums, use wire to clamp the duct inner liner to the metal boot, and cover the seam with mastic.
Look for openings in the connections and corners of the plenums (see pictures). The primary difficulty is often to gain assess to these areas.
Duct boots and elbows
Pay also special attention to the plenum joints at the air handler, and in the adjustable elbows.
They should be carefully fasten and sealed. Use fabric mesh tape if necessary.