Fortnite

EDIT 2: Please go down to JDuke705’s solution below and give him a thumbs up for his amazing trick to find Play With Other missions that you actually want to play(like rescue the survivors)

EDIT: Thanks to this wonderful site: https://www.fortnitedb.com we can track our stats! According to this, when I disconnected from a game and rejoined it did not count towards this. But it did count the mist monsters, survivors, structures built, and zone explored.
Thank you vl3tH4I1tYv for the link!

You can also check your progress via quests under your challenge tab. Thank you Greenway Gaming for finding that!

1000 "play with others" missions. As stated by APOPHIS1989, these are started by using the yellow question mark on your map. You can do these in any area, even if you’re a much higher level. You can do this with a party of 4 as well(thanks Tenshi3).

I've been pretty good at averaging 20 minutes per mission. This includes the 10 minute(sometimes less if you're lucky) storm shield defenses, and the 30-40 minute missions that happen due to randoms refusing to do the mission. My favorite missions to get besides storm shield defenses are "Defend the Grid", "Rescue the survivors", "Destroy the Encampments", and "Build the Radar Grid". These are the best in my opinion because no matter what the randoms are doing, the mission will end when the timer is up! With a coordinated group, exploring the map fully, saving survivors, and doing the mission can easily be done in less than 20 minutes for any mission. Have one person build defenses while the other 3 explore. I recommend staying in Plankerton because once you hit power level 40, this place provides a perfect balance of difficulty with number of mist monsters to kill.

Let's do the math:
Keeping an average of 20 minutes per session means this achievement will take you a whopping 333 hours! In order to get the 1500 zones explored it'll take 500 hours(and it'll take even more than that since any storm shield defense missions you get, you can't explore the zone).

Keep in mind:
Remember to build 500,000 structures while doing these missions. So try and place things down as often as you can.
Hope this helps!

If you ONLY need PWO than you can do the same thing below but with Resupply or Eliminate and Collect missions as they are the fastest at 10 minutes. If you have a boosting partner that has endurance and does not need this you can have them create an endurance and then search for assistance. You can have them join with the Search for storm shield assistance and it will count. Complete wave 1 of the endurance which should be really easy and then leave with party and will count. We tested this with Greenway Gaming.

This will be a long guide with multiple tips throughout. Even with this the achivement is VERY LONG. There is no way to get this quick. This guide is best when working on ALL 3 achievements Save 10k, 1k play with others, and 1.5k full explored maps. This method is easily saving me 100 hours and will save more if you are further behind when starting. The 500k building has a great guide but even with this we average 1200 a match for building. You can check your progrss on these in your Quest log challenges.

At the time of writing I am at
145k/500k=29% buildings
3300/10k=33% survivors
966/1500=64% full explored
5636/20k=28% mist monster
658/1000=66% play with others

Throughout all of Plankerton and Canny Valley I did ONLY play with others to move my quest. Obviously, this was slow and got a lot of explored maps and play with others the other achievements did not move nearly as much. The best mission is rescue the survivors but it hardly comes up with play with others. Also, there are many times in play with others you get people that don't work on the mission, afk, or won't build, or are underleveled which then causes the mission to fail and a waste of 20-45 minutes that is a complete waste since all things have to be on a successful mission.

Play with others will not let you see what mission you are playing until you load in which takes a long time since the game is slow loading. What I have been doing to ensure that every play with others is a rescue the survivor mission is to find a survivor mission in Stonewood. Look through the Rescue the survivors until you find one in progress and take note of who is in it and what power level it is. Use the Play with others and do a custom search for that power level range. If it isn't the person you found before then backout and use the PWO again until you find them. This will require a game to be in session so sometimes you might take a few extra minutes to find the right match or might need to look at another area like plankerton to find a rescue mission but depending on where you are the time that you save by having every PWO mission a rescue the survivor mission will greatly outweigh the extra time searching.

Tips:
Play in Stonewood as much as possible. Preferably not the staring level 5 Rescue if possible but sometimes not an option based on the random maps of the day. This is because the harder levels have more chance that someone lets a survivor die. In Plankerton you have the empty shelters that sometimes glitch and will not spawn a survivor. In Canney and higher you have the towers that sometimes do not spawn a survivor.

If you have a boosting partner switch servers in the settings to Middle East as they are way less busy than others. You can have one person start a match and sit in lobby. Have the other Person use PWO until they connect to lobby. Have the first person back out and then use PWO to connect back. This should get progress for both.

If you have a person with a dummy account this is a godsend. I have an alt that I do not care about achievements on. With this you can start a match when there is 5 minutes left in the first game and load into the game asap. The other game with the primary should finish and you can then use PWO to get into the second game in progress at about the 15 minute mark. If you get past about 13:30 you will not be able to find the game so you can not do this too early so the 5 minute mark is best. This should save 5 minutes PER rescue game. 15 minutes is more than enough time to solo rescue all survivors. Occasionally, you will have randoms join and you can restart the match with the alt account but MOST games I was able to get no randoms and save 5 minutes.

If you have 2 people plus the dummy account you can also make great progress on building. Greenway Gaming and I have been averaging about 1200 objects per game. After rescuing all 15 survivors in Stonewood we spent the last 5-6 minutes building and farming. It helps to have some resources stored up which means you can do some missions in between like resupply to get some resources in your storage and bring about 2.5k of each per match with you. You can also send your expeditions out to get resources. (Thanks Greenwood for the reminder)

Edit: great getting downvote with no explanation. But I can verify this works. In 2 weeks of playing off and on I am now at the following. I will be done with survivor before my 1500 explorer missions. My mist monster kills are coming from idle boosting waves 1-11 of endurance mode in second area. Only time I have to do anything is the UFO round. Once done with these 4 will do building in home base.

As of current (5/14/2020) I am at
Achievement Goal Current Percent
Builder 500000 275352 55.07%
Survivor 10000 6918 69.18%
Explorer 1500 1257 83.80%
Mist Monsters 20000 9155 45.78%
Play W/Other 1000 939 93.90%

To add on to what thepack1221 said, I find doing low level(1-3) custom matches in stonewood to be the best(strictly for this achievement). Almost every time it is a low level Storm Shield Mission. If it's not just remember the name of who you saw in the lobby and back out to try again.

Doing this I can get 1 mission done per 5-7 minutes(This is including load times/search times) + the mini-llamas. Most people have their shield in a box so using a constructor with the B.A.S.E ability pretty much means you can plop it down and just go AFK/tab out.