Definitely use a sport wash on a speed suit, as it'll break down the funk that accumulates on the lycra (so many speed suits smell like they've lived in Animal House).
I wash a lot of tech fabrics (speed suits and higher-end road cycling stuff), so I've used a lot of tech detergents over the years. My faves: WIN (developed by the U.S. Olympic Training Center), Rockin Green Platinum Series Active Wear, and DeFunkify. The first one is a liquid, the other two are powders. For the powders, I find they work better if you dissolve them in a bit of cold water before loading your washables. Go for the unscented versions of these detergents, as the perfumes in these detergents tend to be very strong.
I've used the more commonly available Tide Sport with Febreze, and it didn't do well by my tach fabrics. It tended to set in the funk.
To keep the fabric tight and as pill-free as possible, wash the speed suit inside out. Wash with cold water wash and rinse (hot will set in odors) on delicate cycle. Hand wash if you're worried about how gentle (or not gentle) your washing machine is. Wash it alone, or with other soft tech fabrics. Make sure the zippers are zipped up.
And always hang dry - never put a speed suit in the dryer, it'll get thrashed, stretched, and beaten to death.
And before you put it away for the season, give it a double wash.