What are some good ways to transition to more complex strategic thinking games?
In my experience, the order in which you introduce games is only of minor importance in comparison with understanding what kind of person you're dealing with and what games that type of person would like. So the key is introducing different types of games to identify the gaming personality type. Here are a few gaming personality types I've run across, and the progression that makes sense for them:


Nature-born game lovers. My son is one of these. He had mastered
the strategy of Sorry by the time he was 3 and by the time he was 4
wanted us to teach him every game we had in the house. He started
learning the card game of bridge at age 5. Progression: Order doesn't matter for people like him except for games that require reading and truly complex games like Bridge. He was not a fluent reader until age 7 but some games (i.e. Settlers) could be played anyway by memorizing the pictures on the card. More important than complexity is how interesting the games are. Nature-born game lovers usually want to keep learning new games and they end up getting bored with games of limited complexity and/or limited variation.
Socialites. You may think they hate games altogether but if you
find social enough games ("party" games) they'll play. Examples: Charades,
Pictionary, and many card games. It is interesting that many of
these people end up playing hard card games (i.e. Bridge) just
because it is easy to socialize around the games. But don't expect
these kinds of people to ever play Chess, Go, or any other
complicated game which limits socializing. Progression: Charades is a really nice one to start with. Kids will need the subject area(s) limited to things they know about. Most kids like to play the alphabet game on long car rides  (find a word you can see that begins with letter "a", then "b", and so on). There are other games like this with no components but as they get older you can introduce kids to games like Clue or Squint. Pictionary and some other similar trivia games are simple to learn and play but require extensive world knowledge so are going to be more appropriate for high school age and up.
Game haters. If you try a variety of different games on a person
and he or she hates them all, then give up. Some people just don't
care for games. At best, they may be willing to do 1 or 2 games they
learned as a kid and that's it. Progression: If they truly hate all games, then nothing you introduce will matter. However, be sure to have them try several completely different types of games, in case they only like one specific type of game.
Nature-born engineers. I have a friend who loves games but it
was looking like his son was in "Game hater" category. I gifted them
a copy of Settlers of Catan and we were all surprised that the son
loved it after a history of getting bored with every other game
tried. 2 months later, Eurorails become his favorite game. This kid loves building things so I'm guessing he would like
other games too where you see a network of pieces growing over a
board. Progression: Blokus is a good first one given how simple the rules are. Ticket to Ride is also fairly simple. Settlers and Eurorails are only slightly more complex. I'm not not sure if an empire-building game with economics such as Eclipse would eventually be appealing, as I don't know many people in this category.
Ordinary people (that don't fall into any of these extremes). I
have encountered a surprisingly small number of people who like
games a little but not a lot. Most people I know fall into the first
3 categories. With ordinary folks, I find that if you push hard to
get them to try many new games, or push games on them that take more
than about 90 minutes to play, or ramp up complication too fast,
then they end up not wanting to play at all. So just let them
proceed at a leisurely pace, playing the games they really enjoy. Progression: I would start with very easy traditional games like tic-tac-toe, dots, hangman, checkers, simple card games. You may be able to get them more interested in playing a game to fill in time like waiting for food at a restaurant (which is why I mentioned the first 3 games).
Word smithers. These people love word games and little else. Progression: This is far more age-dependent than any other category. For kids: Hangman, Boggle, Jotto, 25 words or less, Scrabble, might be a good progression. High school and older who already like crossword puzzles can jump right into scrabble.
Strategy-only. These people don't see the point in playing games
with random elements. They will play Checkers, Chess, Go, and
possibly logic and hidden information games like Mastermind or
Stratego. They may play them well - maybe even extremely well. But
the vast majority of games hold little interest due to the luck
element. Perhaps also in this category are people who tolerate modest amounts of luck so long as skill matters the most (Examples: Bridge and Poker). Progression: Tic-tac-toe, checkers, chess is a Western approach. Tic-tac-toe, Reversi (Othello), Go is an Eastern approach. Or a card track could look like solitaire, spades and/or hearts, bridge.
Other. Like the nature-born engineer type I described above, there are certainly other people out there who enjoy only one very specific type of game. The path to discovery is to experiment with very different types of games, and not necessarily with simple ones first.


Certain games are popular because they work well for at least 3 out of first 7 of the above categories of people. For examples, Monopoly, Ticket to Ride, and Settlers of Catan all work well for 1, 2, and 4. I have played a ton of Settlers over the past decade because it offers a particularly good balance for people in these three categories. (Our kid learned it before he could read. First we played without development cards, then he just memorized the pictures on the cards.)

It is likely I missed some types of people - if I did, please mention them in the comments.