Pauper is an Eternal format, meaning the cards never rotate. You can pick up a favorite deck and play that deck forever. Once you have your deck or decks, the only things you need to do to keep up with the format are to watch the card pool by following as new cards are printed and checking up now and then on the metagame. While decklists change, archetypes rarely do. So, once you buy into the format staples, you can keep tuning your new deck to the new technology either as innovators strike new gold, or as occasionally happens, the Banned and Restricted List either removes an oppressively powerful card or opens up a new strategy with the removal of said card.
When I'm building Pauper decks, the format is so cheap that I like to think about acquiring the building blocks for archetypes rather than just the individual cards for decks. Many of the best cards, especially the more scarce and powerful sideboard staples, can slide into a number of decks. So that once you have your playset, you are ready with the building blocks you need for any format deck as it arises in popularity or just piques your interest. Most of these cards are cheap: the vast majority are under a dollar in paper with a couple of the more "choke point" cards that have seen fewer printings running in the $2-$5 range, very few more in the $5-$10 range, and only one card I know of being over $10
So if I were going to start building up my Pauper collection, here is where I would begin:
Preordain is the premier card selection tool in all of Pauper and the single most played card in the format. Since shuffle effects come at a premium in Pauper, it usually takes precedence over the Legacy staple Brainstorm as the best way to dig deeper, to find what you need, and to skip over what you don't. It it's also an excellent tool to stack your draws, say to flip a Delver of Secrets, or to ensure that a spell that you want is the top card of your library to draw for free with Augur of Bolas. Combo decks, tempo decks, and control decks all eagerly jam Preordain, usually the full four.
Played in: Mono-Blue Delver, Izzet Delver, Dimir Alchemy, Tireless Tribe Combo, Izzet Blitz
Pyroblast and Hydroblast are the two most played sideboard staples in the whole Pauper format. Because Blue and Red are not only the most popular and powerful colors in Pauper, but they also come equipped with the best tools to fight each other, these two color pie breaking silver bullets find their way into nearly any deck running either basic Mountains or Islands and occasionally even decks that don't!
Played in: Izzet Delver, Boros Monarch, Burn, Affinity, Tron, Izzet Blitz, Tireless Tribe Combo, Dimir Alchemy, Mono-Blue Delver, and occasionally Elves.
Lightning Bolt is the premier removal spell for any deck running Mountains. It is extraordinarily rare for any deck with access to Red mana to not use its first four spell slots on Lightning Bolts because they are such efficient instant speed removal spells that can also double as reach.
Played in: Izzet Delver, Izzet Blitz, Burn, Red Deck Wins, Boros Monarch
Counterspell is legal in Pauper! While Modern decks may be forced to run pale substitutes, you can feel all the joy of overshadowing your opponent's turn by leaving up two untapped Islands.
Played in: Izzet Delver, Mono-Blue Delver, Dimir Alchemy
Ponder is the second best card selection spell in Pauper and trades the ability to "bottom" cards with the ability to shuffle your deck for no additional mana. Not only does this spell give Blue decks all the usual card selection business of stacking your deck, but it also sets up trading dead cards for fresh draws in combination with Brainstorm.
Played in: Izzet Delver, Mono-Blue Delver, Izzet Blitz, Inside Out Combo
Prophetic Prism doubles as both the color fixing cog that converts the copious colorless mana of Tron decks into powerful colored spells and also the cantrip engine of the Boros Monarch deck. Both these decks tend to shift and evolve with time, but the playset of Prophetic Prisms remain indispensable pieces to each of these archetypes.
Played in: Boros Monarch, Affinity, Tron
Gush is one of the most unfair draw engines imaginable. Both combo decks and "fair" decks use it as a free draw spell that can be doubled as a way of either trading excess lands for more cards or to refill the hand to fuel Tireless Tribe. This is a big card to purchase, but it is one of the most broken cards you can play in Pauper.
Played in: Mono-Blue Delver, Izzet Delver, Izzet Blitz, Tireless Tribe Combo
Gorilla Shaman is the best hate card in Pauper. Much like Dredge can be the bogeyman of older eternal formats like Legacy and Vintage, Affinity is a deck that can destroy a metagame if their opponent's choose to skimp on hate. Gorilla Shaman is not only the best hate for this powerful aggro deck, but it gains collateral advantage in the Boros Monarch matchup too.
Played in: Izzet Delver, Izzet Blitz, Boros Monarch, Tron, Affinity
Relic of Progenitus is another amazing sideboard hate card that specializes in graveyard control. The opportunity cost of adding this to your sideboard is so low that almost any deck can run it. Its failsafe scenario is that it even cycles from play for two. But the added value of your first graveyard hate card to shut down recursive loops like Ghostly Flicker or Tortured Existence is so low that it is the first universal sideboard card to add to any collection.
Played in: Boros Monarch, Elves, Affinity, Izzet Blitz, and nearly every rogue deck.
Dispel is a premier sideboard staple to win counter wars, stop combo decks, or even just eat to eat a Lightning Bolt. And the best part: a playset will cost you less than a dollar!
Played in: Izzet Delver, Izzet Blitz, Inside Out, Mono-Blue Delver Tron, Affinity
Mulldrifter is not only one of the most fun cards you could cast, but the best creature for big mana decks. It gives you a 3-for-1 for the full , or conveniently will let you play it as a Divination for . It makes a fantastic loop card with Ghostly Flicker, and loves to be recurred from the graveyard via Pulse of Murasa or Soul Manipulation.
Played in: Dimir Alchemy, Tron
Gut Shot is the Burn spell of choice for any deck without easy access to Red mana. It is an excellent way to pick off Delver of Secrets, Quirion Ranger or can kill a Spellstutter Sprite with its trigger on the stack to safely resolve another one mana spell.
Played in: Izzet Blitz,Tron, Mono-Blue Delver, Elves, Stompy, Bogles, Tireless Tribe Combo
Gitaxian Probe is another Phyrexian mana spell that, like Gut Shot, has a habit of being played for zero mana and 2 life a lot more than its colored mana cost. It is used to get information for combo decks, add triggers for spells-matter cards like Kiln Fiend, and most recently, has found a home in Burn decks as a way to turn on Ghitu Lavarunner a turn early.
Played in: Tireless Tribe Combo, Izzet Blitz, Affinity, Burn
Augur of Bolas is the single most played creature in Pauper. In a spell heavy deck, it will blindly draw a spell north of 70% of the time, and it is trivially easy to stack the top of your deck to ensure a specific spell drawn with cards like Preordain, Brainstorm, and Ponder.
Played in: Izzet Delver, Izzet Blitz, Mono-Blue Delver, Inside Out Combo, Dimir Alchemy
Delver of Secrets is the default best threat in Pauper, and gives the deck its namesake tempo threat, much like it does for its Eternal deck cousins.
Played in: Izzet Delver, Izzet Blitz, Mono-Blue Delver
Electrickery is Pauper's best sweeper. At instant speed, and with a built-in flexible mana cost, it it picks off opposing X/1s while leaving all yours untouched. It brutalizes the Elves deck, is a necessary weapon against both Spellstutter Sprites and tokens alike, and is one of the only ways to successfully kill hexproof creatures like Slippery Bogle and Silhana Ledgewalker.
Played in: Izzet Delver, Izzet Blitz, Boros Monarch, Tron, Affinity
Chainer's Edict is the premier removal spell in Glack-based control strategies, but it is also one of Pauper's most expensive cards. Because it only ever had a single paper printing at uncommon plus a couple of special set foil printings, copies now push $10 each, which can be hard to swallow for a common. At least it's not Oubliette.
Played in: Mono-Black Control, Dimir Alchemy
Quirion Ranger and Nettle Sentinel are the bread and butter Green common creatures and the overlap between the two best Green decks in Pauper, Stompy and Elves. Quirion Ranger does great tricks like resetting lands for extra mana, giving attackers pseudo-vigilance, and letting creatures with activated abilities use their abilities twice in one turn. You can even activate Quirion Ranger on both your turn and your opponent's turn if you have two Forests in play.
Played in: Stompy and Elves.
These are the first twenty cards I would buy if I were going to build my Pauper collection. Yes, there are a lot of Blue cards and zero White cards, but that is the reality of Pauper: the best Blue cards go in far more diverse decks. I tried to feature the cards that fit in the most different decks so that you can identify the flexible cards that you can buy once and port into any new deck you build. Happy deck-building!