Exactly what dances you begin with often depends on what's being taught when you're available. Look for nearby dance studios and find out what classes they're offering. If you ask the studio, they will surely be able to recommend where to start.
Some dance studios offer beginner classes that cover multiple different dances at once. This is a great way to start out because it gives you a chance to experience some of the variety that exists, plus it gives you a wide base to begin with and dance multiple dances right away. When attending ballroom dance parties (which are a great way to practice beyond classes), they usually play a variety of music for all the different dances, so knowing more than one dance can give you more opportunities to dance. And as you experience the different dances, you will get an idea of which ones you like and want to pursue more.
There are a variety of different styles/categories/branches of dances, and each of them have a good dance to start with. A simple list of dances in different categories that I would recommend starting with would be: Waltz, Rumba, Nightclub (Slow) Two Step, and Salsa. Each of these provide a good and simple base to start with and build from as you progress to other dances in the same category.
There is no "right" or perfect order in which to learn any of the dances, and as you go beyond the basics, you will likely find your own reasons for learning more dances in the order that you choose. But for the purpose of offering some idea, this would be my recommended order of progression in three of the major branches of dances:
Waltz, Foxtrot, Tango, Viennese Waltz, Quickstep.
Rumba, Cha-Cha, East Coast Swing, Mambo, Bolero.
Nightclub Two Step, Hustle, West Coast Swing.
One other factor that could greatly influence the dances that you learn and the order that you learn them could be where you want to do your dancing. If there's a club that does Latin dancing, you might end up learning Merengue, Salsa, and Bachata. If you want to dance to more typical club/pop music, you might focus on West Coast Swing and Nightclub Two Step. If there's a big swing dancing crowd in your area, you might stick with East Coast Swing, Jitterbug, Lindy, and Balboa. You could even spend all of your time learning nothing but Argentine Tango. Then there's all of what I listed above if you're interested in traditional ballroom dance parties. And if you're more interested in dancing to compete rather than social dancing, that can also focus which dances you learn and in which order.
All of these options may seem overwhelming, but it all comes down to personal preference and what you enjoy in your area. My most basic recommendation is to find a place where people are dancing something that you're interested in, and ask them how to get started. Most of the people I have met through the ballroom dance community are very friendly towards beginners and are eager to help them get further into dancing.