I got into board-gaming about 10 years ago and started off with one of the heaviest euros immediately - Agricola. It has since remained one of my favourites with its endless replayability. Other classic euros I like or love are Caylus, Power Grid and Puerto Rico.
Soon after Agricola I landed in the world of Vlaada Chvatil's games. His Through the Ages - A Story of Civilzation (and its recent new edition) and Dungeon Lords are two games I absolutely love. Dungeon Petz and Codenames are quite good too. The only game of his I have a hard time with is one of his most lauded - Mage Knight.
I tried Descent several years ago, but never liked it. Now after I've tried RPGs for the first time we recently tried Descent (2nd edition) again and I've come to love it. Compared to our RPG sessions it's more fast-paced, more structured and less grinding, with all players feeling like they have something to contribute.
In the cooperative realm we started with Battlestar Galactica, but soon learnt that it truly only shines when you have exactly 5 players. Nowadays it seldom comes to the table. Basic Pandemic was also tried some times with some other friends. 1.5 years ago someone in a Facebook group invited me to join Pandemic Legacy, and 4 of us ripped through it in 2 weeks in the most immersive gaming I've ever had. I then bought it and played through it a second time with my regular game group. I refrained from contributing to strategic decisions and had a lot of fun seeing them experience the story. I've even gone so far that I've watched several groups play through it on Youtube (most notably Dice Tower). I've since bought both Pandemic - Reign of Cthulhu and Pandemic Iberia. Other co-ops I love are the somewhat more stressful and less prone to alpha-gaming: Space Alert (by Vlaada Chvatil) and XCOM. Especially XCOM can really get your heart pumping. On the lighter side Mechs vs Minions was released last year at a (for its content) low price. I'm going to have my first go at miniature-painting with it.
I only tried the gateway games after several years of gaming but I really like both Ticket to Ride and Carcassonne (both have been played hundreds of times on my phone). I've successfully introduced TtR to several family members. Settlers of Catan, however, I only play 1 or 2 times per year. It can be fun in limited doses.
I'm looking forward to trying Time Stories, Great Western Trail, some more XCOM and Descent, and Pandemic Legacy Season 2 and First Martians in the autumn.