When I got started I got queasy and sick for three reasons as far as I can figure.

First, I sometimes got green tobacco in my cigars (Thanks, General Cigar, more specifically Punch). You'll know if you get green tobacco, as soon as you smoke the green tobacco part of the cigar you'll get a massive nicotine infusion, you'll likely get a iron-spike-in-the head headache and get very queasy.

Second, you may be smoking them too fast. Slow down, pace yourself, the slower you smoke the more even the smoke will be, and in my experience, the more flavor the cigar will have.

Third, make sure you have some food on your stomach before smoking strong cigars. Any good retailer can tell you if you're buying strong smokes or not, and if they can't, don't go back to that cigar shop, they don't know shit and don't deserve your business.