Do you have any recommendation from your experience/knowledge?
Then I would say look at styles and research the instructors. I guess style research could come first if there are obvious things you don't want, like for example if you don't want to practice something with lots of ground work that is like wrestling, then avoid BJJ/MMA schools.
It can be expensive but when I went (long time ago) I did find it to be a good workout where I wasn't bored, like in a gym.
I'd also suggest finding a school that has emphasis on sparring and self-defense. Yes, the martial art is interesting, but at the end of the day it's practical to learn good street fighting techniques. Your martial art by itself isn't all that good for a fight involving guns, knives, baseball bats, and the like.
It's also good to have some ground-fighting techniques. You don't need to be a BJJ whiz, but you also don't want to be a fish out of water if you get taken to the ground. My TKD class was taught by a fireman, and it was 70% focused on TKD, about 20% on self-defense, and 10% on ground-fighting.
The school and the instructor are more important than the art itself.
But it's not for everyone. I'm trying to get my partner to try taekwondo with me, for example, because the teacher there focuses more on basic body mechanics (this is how you throw a punch, this is how you kick) whereas a lot of muay thai places jump right into partner drills with insufficient focus on basic form, IMO.
Brazilian jiu-jitsu is a natural counterpart to muay thai (since both are used in MMA, and most gyms will teach both). It's very good strength & groundwork from what I understand, though it's not something I'm familiar with (too much rolling around in other people's sweat).
But I'd really recommend trying several studios and styles (at least a few days at each one, preferably a few weeks) to see which one suits your needs the best. The teacher & studio can make a big difference.
Examples of things to consider:
* Is this class giving me enough of a workout? Is it all cardio or is there some strength & core mixed in?
* Is the class giving enough of a challenge for different ages & ability levels, or am I mostly sitting around while kids flail about?
* Does the teacher drill the basics competently, as opposed to just throwing me at the bags or partners and letting me loose?
* Is there a culture of safety and mutual respect? Does anyone seem overly aggressive or dangerous? That should NOT be the case in your studio; the potential for injury is very real if your partner(s) are dicks.
* (Least important at the start, IMO): Is this the style I really want? Does it offer the right combination of striking/grappling/throws/forms/self-defense/fitness I'm looking for? I say least important because at the beginning, it's way more important to get your body used to the basic mechanics (shared between many stand-up martial arts) than to find your perfect match. You can always switch later, once you develop the basic knowhow and better know what you like and don't like. It's not immediately obvious at first.
If I were starting out new (or getting someone to try one for the first time), I'd probably look for the crowded adults-only classes around me, watch or join them a few times (most places have free trials) and see how they go. Style IMO is less important than class structure, until you're sufficiently experienced that you develop preferences.
If you're really just after physical fitness and self-discipline, a martial art may not be the best solution. Maybe yoga, acrobatics, or some forms of dance training. If you really want a martial art but aren't interested in combat, consider Tai Chi. The techniques do have combat applications but very few instructors teach it that way. The emphasis is usually on fitness, balance, bodily harmony, and a calm mind.
I'm no expert but I did four years of (very) traditional Okinawan karate and three more of a full-contact non-traditional "mixed" style called Jeet Kune Do, plus limited practice in other styles via seminars and training exchanges with different schools.
Schools teaching classic Asian arts like Karate, Kung Fu, Tae Kwon Do, etc tend to incorporate Asian culture and teaching methods as part of the package. A lot of people in that world are drawn to it because of that Asian-ness. If you're after discipline, regimentation, attention to exact form, and immersion in Asian history, that's where you'll find it. Emphasis on fighting effectiveness varies widely, as does the mood of various schools. One school may seem like it's full of bullies and another teaching exactly the same style will feel welcoming and supportive.
But don't count out the more combat-oriented arts. Krav Maga, for example, was originally developed as a hand-to-hand fighting system for the Israeli army and still keeps that spirit in mind, but there's a heavy emphasis on teaching and students' personal development. There definitely is contact but they're careful not to throw you into situations you're not ready for. You may find that becoming gradually acclimated to that physical rough-and-tumble in a safe enviroment (apart from the fighting per se) can transform your sense of self-confidence.
All martial arts training, including familiar western forms like boxing and wrestling, build self-discipline and personal development. It may be more tangible and visible in the Asian arts because of how they're taught (standing in lines, standardized exercises, ritualized greetings, etc) but it's there, just not as performative.
As others have said, the experience can vary hugely from one school and one instructor to the next. My advice would be to look at a number of styles and see if one of them resonates with you in a non-intellectual, intuitive way. One approach is to find seminars or exhibitions where practitioners of multiple styles do demonstrations and short instructional sessions. Trust your feelings more than your head. Wear comfortable gym clothes you can move and sweat in and go see what it's like.