A selection of board games to try:
Shogun (2+ players, best experience with 4 players, strategy)
One of the best grand strategy games I've seen, it has a random element with global events, during the battles and selecting provinces which I find not too disruptive. Most of the game is focused on action allocation, both military and economic decisions which translate to victory points after each of the game years.
Fury of Dracula (2+ players, best experience with 3+ players, action/roleplaying elements)
This is a very enjoyable game following the standard one player with all others cooperating against him. The antagonist plays as Count Dracula and depending on the time of the day, has a wide range of abilities to disrupt opponents, travel or interact with the location by setting traps and roadblocks. The goal of the vampire hunters is to defeat him, while Dracula has to create vampires using various game mechanics. One of the most enjoyable features is the track left by Dracula, that lasts several days and can be discovered by the hunters while they encounter the traps.
Twilight Struggle (2 players only, the best 1v1 experience in board games, strategy)
The description can be found online, this game simulates the cold war struggle between the US and USSR and its only negative lies in being a two player game.