Axis and Allies can be a great intro to heavier wargames, but definitely not a “family game night” sort of game.
Risk is… I do not like Risk.
With five players, try The Resistance. Social deduction / bluffing, with no player elimination.
Dead of Winter is also a great play. Co-op. Sorta.
Second this although it is very expensive to buy for a board game.
Still, lots of fun. Here is the gameplay on TableTop with Wil Wheaton.
I played that this week. We almost lost when somebody got bit entering the compound and infected two other people. Our morale dropped down to one but we were able to ride it out.
One thing that I can add about these coop games - Pandemic (including the legacy version), Eldritch Horror, Arkham Horror, etc - is that you are sometimes going to lose. Play them for the journey, not the destination.
They played Flash Point - Fire Rescue on the episode of Tabletop that released to YouTube this week. It is a co-op game that can be played with 2-6 players. I plan on picking up a copy at my game store this weekend.
Not to discourage anyone from patronizing their local game store but there is a current Kickstarter campaign for a new Flash Point expansion. If you back the expansion, you can also choose to buy the original game and the other expansions as an add-on. Or you could buy the original game at your local store and then choose to buy the Kickstarter package which includes all the expansions but not the original game.
Codename: Pictures is a great game for 4+ people that works fine with large groups (no problem with 8-10). You lay out 20 cards on the table and draw one layout card that assigns colors to each card. Each team picks a ‘caller’, who is the only one who can see the layout card. The callers alternate giving one-word (plus number of cards) clues, while the rest of the team tries to figure out which cards to flip over based on the clue; you only want to flip your own cards, and want to be the first to flip all of them. It’s really quick and simple to explain, it’s very unlike something like dominion where the base rules are fairly complicated and the cards are effectively a bunch of other rules to track each game.
There are a bunch of variants, but the picture one is my favorite.
If you have Codenames and Codenames: Pictures try shuffling the decks together and playing the game with a combination of words and images.
Risk was always the “go to” game among my crowd. If you’re playing with family, though, you have to say and mean that all players are equal and when the game is over, it’s over. When my buddy’s sons got old enough to play, cries of “You suck, Dad” were not uncommon.
Amazon is having a one day board game sale right now.
Takenoko is the best game that I have experience with that is on the list.
From that sale, Citadels is a classic game and plays very well with five players.