What are the correct Jewish procedures for finding someone to marry? Like does "dating" have any Jewish connection?
The requirement for "Shidduchei"
The Gemara (Yevamos 52a) says that "rav menagid ... demikadesh b'lo shiddhuchei", a person deserves to be whipped if he gets married in a crass, inappropriate way, without having shidduchei first. Some might interpret that to mean there has to be an official formal matchmaking document or process or something before the marriage, but what I've heard from both Rabbi Aaron Rothkoff-Rakefet and Rabbi J David Bleich is that the Gemara simply means "getting married without whatever courtship process is normal in that culture." (Rabbi Rakefet suggested, for example, meeting the prospective spouse's parents; Rabbi Bleich suggested if the couple worked together on any joint planning of the wedding, that qualifies.)
The requirement for a man to see his prospective wife before marriage
The importance of loving one's spouse
The Maharik (Italy, 1400s), 166:3 was asked regarding a guy who had fallen in love with a young lady, but his father objected. Maharik observed that the mitzva of honoring parents does not apply if the parent says to go against the Torah. As one is required to love their spouse, the father's request to dump this lady and find someone else (who he wouldn't love) is of no halachic standing.
Dating for marriage, vs. for fun
Different streams of observant Jews have different senses as to how much social contact between teenagers of opposite genders is appropriate. But R' Moshe Feinstein writes that a teenage boy really shouldn't have a serious "girlfriend" if this has nothing to do whatsoever with marriage.
Even if something per se isn't a mitzva, if it enables a mitzva to happen, that's important too. What sort of dating could then be considered "enabling" a happy marriage? I leave that to you to decide.
So what does this all mean?
Again, the only black-and-white requirements listed are:
How does that translate into practice? In some communities, the parents arrange everything, the couple sees each other once, and if they think it works, that's it, mazeltov. In other communities, singles meet on their own, and tell their parents shortly before the official engagement. In many communities, it's somewhere in between, which can be vague.
(Note the Gemara at the end of Taanis sounds like there was a singles population that met at marriage-minded-singles events and worked it out on their own, though I recall that the (7th) Lubavitcher Rebbe zt'l interprets that Gemara non-literally, as a relationship between God and the Jews.)
Certainly if someone feels that s/he're not getting enough dating to cause the likelihood of a happy marriage, s/he should talk with his/her rabbi. Rabbi Moshe Bick wanted to require a minimum of eight dates before engagement, even in the Hassidic community. (He'd seen too many broken marriages.) Rabbi Yoel Teitelbaum of Satmar bristled at the idea.
Lastly, even within a community, there may not be a one-size-fits-all approach.