TL;DR: The survivors get water a number of ways, mostly depending on their location.
Ponds, creeks, and quarries
A bottled water delivery truck
A fire truck
Rainwater collected in several different ways
Desalinated seawater (on Fear the Walking Dead)
As for your comment:
Access to water never seems to be a problem for the survivors.
This isn't true. It was a major problem in the first half of season five, and has been mentioned at several other points on the show.
The Walking Dead:
While the survivors were camping at a quarry on the outskirts of Atlanta, they boiled water from the quarry for drinking and sanitation:
Shane drives back to the camp with water
Shane hauling water from the quarry to the camp
Shane: Water's here, y'all. Just a reminder to boil before use.
- The Walking Dead, Season 1, Episode 3: Tell it to the Frogs
Shane goofing around with Ed at the quarry
Quarry campsite filming location on Google Maps - note all the water
As you might expect, Herschel's farm has several wells:
Glenn: Miss, what's the water situation here?
Maggie: Got five wells on our land. House draws directly from number one. Number-two well is right over there. We use it for the cattle but it's just as pure. Take what you need. There's a cart and containers in the generator shed behind the house.
- The Walking Dead, Season 2, Episode 4: Cherokee Rose
Unbeknownst to Maggie, the well she tells Glenn to use is not just as pure as the others, as Dale and T-Dog soon discover:
T-Dog pumping water to slake his thirst while Dale snoops around a bit
This is the well version of the world's worst Cracker Jacks prize
But, as Maggie said, there are four other wells that don't have rotting zombies in them, so the group uses those for the duration of their stay on the farm.
By the premiere of season four, the prison has been adapted to support a fairly large community of survivors. Water is drawn by a hand-pump from a pond and small creek just outside the fences. The pond and creek are rather gross and muddy, so the survivors must have boiled their water before drinking it.
Prison exterior, with the pond in the foreground
Carol clears a clog in the intake hose filter
Carol operating the hand-pump while Rick just stands there watching
The prison set was actually the back lot of Raleigh Studios in Senoia, Georgia1, and the pond and creek (called "Keg Creek" in real life) can be seen in this satellite photo from Google Maps:
The prison set - the larger pond on the right is the one where the Governor disposed of Pete's reanimated body
Before the outbreak, the Alexandria property was designed as a fairly expensive, ecologically friendly gated community. As such, it was equipped with cisterns for water collection.
Rick: What is this place?
Deanna: This is the start of sustainability. That's what the brochures we found say. This was a planned community with its own solar grid, cisterns, eco-based sewage filtration. Starting in the low 800,000s... (laughs) .. if there is such a thing.
- The Walking Dead, Season 5, Episode 12: Remember
In a few shots, we can see the cisterns. They are small plastic rain-barrels of the sort you can buy at any big home improvement store, and are filled via a pipe from the gutters on each house:
Rick's hospital room:
When Rick emerges from his coma in the series premiere, the taps are still working:
Just before finding Herschel's farm:
The group is so desperate for water that they try to siphon it from cars while stuck on the road:
I can siphon more fuel from these cars for a start. Maybe some water.
- The Walking Dead, Season 2, Episode 1, What Lies Ahead
And Shane and Glenn are almost euphoric when Shane finds a truck full of massive bottles of water. The looks on their faces shows just how desperate they had been, and how much this discovery means to them:
Shane spots a truck with "Mountain Spring Water" written on the back, and decides to investigate
Shane opens the truck, sees all the jugs of water inside, and steps back, awestruck
Shane: Glenn, were we short on water?
Glenn looks over, sees the water, and his jaw drops
Shane pops the cap off a bottle of water and lets it pour into his mouth
Glenn laughs maniacally
Glenn: Save me some!
Shane lets the water pour over his head
Shane: It's like being baptized, man.
- The Walking Dead, Season 2, Episode 1, What Lies Ahead
When Carol, Tyreese, Lizzy, Meeka, and Judith are on their own after the prison was destroyed, they find a house in a grove of pecan trees, and Tyreese fixes the old water pump in the yard:
Which is a marked improvement over what they had been doing for water:
On the road after Terminus:
The survivors are near death from dehydration. Some of them try digging for water in dry creek beds, to no avail. Then they come across a stash of bottled water in the road with a note reading "from a friend".
As desperate as the group is, they are also understandably suspicious, having just escaped a community of cannibals who lured them in with promises of safety, then tried to eat them. Only Eugene is willing to drink the water; Abraham, however, slaps the bottle from his hand before he can do so.
Just then, their luck changes, and they make good use of a sudden, torrential downpour.
In episode six of season five, Consumed, we see a flashback to what Carol did after being kicked out of the prison by Rick in season four. She went to a nearby office. We see her rigging some plastic bags outside the window to catch rain:
On the road to DC:
When Rosita, Eugene, Abraham, Tara, and Glenn are on their ill-fated trip to DC, they first find water in the tank of a firetruck:
Later, they find a disgusting shallow creek, and Rosita builds a water filter to remove most of the nastiness:
Tara: So, you just built a water filter in about two minutes, huh?
Rosita: Eugene taught me. Takes a while, but it'll be clear. Then we can boil it.
- The Walking Dead, Season 5, Episode 7: Crossed
Grady Memorial Hospital:
The hospital where Beth is kept captive has at least two large rain collection bins on the roof:
Fear the Walking Dead:
The protagonists on FTWD have been fortunate, compared to the characters on TWD. At first, they were in their homes with running water; then the water stopped flowing, but the military supplied them with bottled water. In season two, they've spent most of their time on a yacht with a desalinization system that makes sea water safe to drink:
Man on radio: Yeah, how about you? You have water?
Alicia: Uh, yeah, we have a desalinization system.
- Fear the Walking Dead, season two, episode one, Monster
1As you can see on my Google Map of TWD filming locations, Senoia is the most frequent filming location for The Walking Dead:
TWD filming locations in Senoia, Georgia