What to recommend for first drama?

So they'll get hooked, a good starting point.We are, however, trying to avoid (sorry for the term) shallow high school main characters

The first Korean drama I watched that got me hooked was Reply 1997, but it is a high school story (though I wouldn't exactly classify the main characters as shallow). Originally when I was not too familiar with the actors/actresses I tended to get them mixed up so I preferred stories which revolve around just a few key people. 

Like 3ngin33r above mentioned it depends on what the person likes (also does the country of origin matter to you?), but here are some of my suggestions for the different genres:
- Comedy: Coffee Prince
- Medical: Emergency Couple or Good Doctor
- Action: Healer
- Melodrama: Innocent Man
- Thriller: Liar Game (or the Korean version, which I haven't watched but I've heard is good)
- Historical: Sungkyunkwan Scandal
- Cop: I Remember You

I agree with 3ngin33r, it depends on what the person likes. But I think the easiest way is to choose something with multiple genres.

For my friends who liked mainly romance, I choose "You who came from the stars" (SF + romantic comedy). Just after one episode of this, which they really liked, I tried to let them discover "Kill Me Heal Me" (romantic comedy + mystery). But this one couldn't get them interested because they had a hard time differentiating the actors. Whereas in YWCFS, it was way easier.
For my parents, after seven years of struggling, I finally succeded in making them watch a whole drama with "I Hear Your Voice" (SF + mystery + romance + comedy). They needed a very good mystery plot with some humour to be interested.

If you want to find a period drama, I would suggest to not choose one set in old time, or at least not fully. Queen In Hyun's man could be a good idea because it's a time-travel drama with many alternations between the two time periods. You can also try Bridal Mask, which is set around 1940 I think. It's not that much into the past but you begin to have the feel of an ancient period drama.

i think it would be better if you shared some more info about yourself. Maybe age or at least you favourite non-asian tv shows/movies?

Personally (and i know this opinion is kinda weird) i think it is better to start with dramas that are a bit more on the 'average' side or at least not with the ones that are rated the highest. the highest rated dramas are usually somehow different than others and i am not really sure if a person who have not seen the 'norm' can be as excited as the rest of the community. (not to mention that even average dramas can be good, especially when you have nothing to compare with it :D)

However! The dramas which were mentioned are good and if you think you might like them then go for it ;)

In my experience people fall for dramas with the most unexpected dramas you could think of, and they are not necessarily the best ones! Most of us started with an awful drama, but it somehow called us right? So just let the person choose based on their likes and the program they use. For example, if they use Netflix then they only have a few choices. 

Obviously, it's Boys Over Flowers! <3

Definitely Fated to love you(korean version)
simply because it has everything typical in a korean drama only better because you
Cannot beat the acting!

People have been recommending Kdrama but since you didn't state the language, I know what NOT to watch as your first Thailand drama: Sawan Biang, Jam Loey Rak. Seriously they've been recommended here and there but if it's your first time try to look somewhere else.