Here are some of the best ArmA III Single and Multiplayer missions.
Single Player Missions:
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Rescue The Doctor: Lovely short mission that makes great use of ambient civilians.
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Village Sweep Altis (SP): A great short mission with control of multiple squads and supports.
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Road to Redemption (SP): A long form mission where you clear a number of approaches and villages along the Kavala Highway.
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[TDNL] Flashpoint - Ambush: A good remake of the original Op Flashpoint mission with the original voice acting.
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Wulfenstein - Level 1 (SP): Although potentially a silly idea, this is actually awesome training for CQB in Arma3
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GhostHawk Shift: An enjoyable single player transport helo mission. Good practice for flight school
Small Group Coop Missions:
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The Great Altis Highway Race (for up to 10 players): A race down the main highway across Altis - really great fun! There are two versions of this a daytime and a night/fog version. Not as serious as other missions, but will keep you and your group entertained!
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Operation Greenstorm (for up to 4 players): A bit laggy in town (due to too many AI), but really nicely organised and very atmospheric - it uses the new volumetric fog.
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Trouble in Paros (Altis) (for up to 4 players): Tough, but good and you get 3 Darter UAVs to play with!
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Strikeback (for up to 3 players): Provide CAS from a Blackfoot, hunting down enemies in and around Kavala. Good fun!
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Hunted Bear (for up to 4 players): A nice simple mission set in and around the cargo storage yard and solar power farm just south of the main air base. Also a good chance to do some sniping!
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Take Kavala (for up to 6 players): Storm the biggest city on the island!
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Operation sunrise ( co-op 02): Really nice 2-player mission, 1 player leads a sniper team and 1 player leads an infantry unit on a good set of tasks. Performs really well in MP. Recommended!
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Sniper in action 1.02 (for up to 2 players): A well-done sniper mission for a sniper and spotter. It is not overly heavy on enemies, but does have some nice elements.