Sugar is the cure. When you get a buzz like that from a strong cigar (or two, or three…), sugar will make you feel much better. The best way is to take a packet or two of sugar, put it on the back of your tongue, and drink a glass of water. It helps quite a bit. Smoking on a full stomach can be helpful as well.
Also, think mild. Cigars with a mild strength profile appeal to many beginners. So what cigars are mild? Some brands that come to mind are Macanudo Café and some Davidoffs. Try searching our online ratings database, too, to find milder cigars. These are all high-quality cigars made with an easygoing blend.
It's easy to make general suggestions, but the only person who can truly judge your taste is you. Try a few different brands, and make note of what you like. If you enjoy the cigars with a mild flavor—cigars that are described as mild in the tasting notes in Cigar Aficionado magazine, for example—then that type of cigar is probably for you. If you don't enjoy that type of cigar, try something in the medium-bodied category.
For starters, you'd probably be wise to avoid anything described as full-flavored, powerful or strong. But enjoy the search—one of the best things about smoking cigars is the process of discovery.