Just like you don't wash your hair for a few days after having it dyed, you shouldn't have your hair relaxed for a minimum of two weeks after dyeing, if not a month or longer. If you hair is commonly dyed and relaxed, you should give it even longer recovery time, or consider a hot oil treatment to regenerate the lost moisture and nutrients in your hair.
The side effects of misuse or too early an application can be your hair falling out in large clumps from chemical overkill. Generally it's anything from hair breakage or hair thinning, through to the lack of hair growth, scalp irritation and scalp damage.
After receiving the treatment, you should keep your hair very moisturised, but not washing it too often; only once or twice a week. Over moisturising will compensate for the chemical impact.
You can use a no ammonia semi-permanent dye on your hair after a relaxer, but it isn't recommended. Going marginally lighter in colour isn't too damaging either, but the lighter you go the more peroxide it uses and the more damage is caused.
The best plan is to consult a professional stylist or hairdresser. They will know how to best get the results you want without causing excessive damage, and how to repair after treatments, should you choose to have your hair relaxed.