Now then.
What you’re asking for is what board gamers term “gateway games.” Basically these are modern games which are accessible to people who haven’t played a board game since, well, Monopoly. Gaming has moved on in the last century or so and there are many, many great games out there, a lot of which are insanely complicated and would put new players off for life.
If you like Monopoly – and are actually playing it properly by auctioning properties – then Settlers of Catan is essentially THE gateway game. It started a revolution. It’s pretty easy to learn and has aspects of trading like Monopoly. You roll dice, collect resource cards based on the dice rolls, then spend those resources to build roads and houses. To get anywhere you need to trade resource cards in your hand with other players. A word of caution here though, Catan requires at least three players, it doesn’t work very well at all with two.
As others have mentioned, Ticket to Ride is a solid shout. You have a map, cards with routes on (eg, London to Madrid), and have to build train lines on the map to complete the routes in your hand. There’s a bewildering number of versions and expansions but all you really need to know is this: There’s two base games which are both compatible with all the expansions, USA and Europe. USA is the original game, Europe its successor. Europe, being the later game, is slightly more complicated than USA with a couple of extra rules to worry about, but also slightly more refined. There’s a couple of special edition base games also, Marklin and the 20th Anniversary box, the latter is beautiful but neither are compatible with the expansion sets so I’d avoid these as a first purchase.
Carcassonne is another classic. You take turns to draw map tiles and then place them against what’s already on the table, lining up features like roads and cities Dominoes-style. You have a handful of little player pieces called Meeple which you optionally place after laying a tile. When you complete a feature such as a city, the Meeple is returned to the player’s hand and they score points based on the completed feature. Simple to learn and much more tactical than it appears at first glance. There’s a shitload of expansions for Carcassonne also, the first couple are excellent additions but beyond that it quickly turned into a cash cow and there’s some pointless tripe out there. But we can revisit that another time if you get to that point.
If you were ever a fan of Risk, you may enjoy Small World. This is an area domination game, you choose a race and a power (eg, Berserk Elves) and go out conquesting the map. Once you’re spread too thin to go any further you put them into decline (essentially, they die off), choose another race / power combo (Flying Wizards!), rinse and repeat. The actual gameplay mechanic is straight forward, the complexity comes with the races and powers which do different things. So for instance if you’re “flying” that means you can attack anywhere on the map rather than having to be adjacent to the rest of your tribe.
My current new favourite gateway game is Sagrada. You make stained glass windows. No, wait, come back! You have a bag with a ****ton of dice, and each player has a 4×5 “window” grid. You draw dice from a pool (called ‘drafting’) and place them in your window according to placement rules – the same number cannot be adjacent, the same colour cannot be adjacent, and the background of the grid has additional rules where for instance a given square has to be red. You get points at the end for completing goals defined by cards drawn at the start of the game. It’s quick, fun, and very pretty.