Pokemon-go – Does Pokémon GO have any interoperability with handheld Pokémon games

As of 2016, it is possible to trade from the Virtual Console editions of Pokémon Red, Blue, Green and Yellow to the new Pokémon Sun/Moon (Gen 7) via Pokémon Bank.

Unfortunately while Gen 6 (X/Y/OR/AS) is compatible with Pokémon Bank, it is not possible to receive Gen 1 Pokémon in those games.

However, if we're talking about the original Gameboy cartridges, it can't be done. The limits on trading are as follows:

Generation 1 (Red/Blue/Green/Yellow)
↑↓ Both directions
Generation 2 (Gold/Silver/Crystal)
X Trade impossible
Generation 3 (Ruby/Sapphire/Emerald/FireRed/LeafGreen)
↓ One-way only, allows held items
Generation 4 (Diamond/Pearl/Platinum/HeartGold/SoulSilver)
↓ One-way only, no items
Generation 5 (Black/White/Black 2/White 2)
↓ One-way only, no items
Generation 6 (X/Y/Omega Ruby/Alpha Sapphire)
↓? (Unreleased, backwards trading yet to be confirmed, Pokémon Bank currently forbids items)
Generation 7 (Sun/Moon)

There are three main reasons you can't trade between Gen I/II and Gen III:

1. The Pokémon data format changed significantly in Gen III, so the two were incompatible.
2. The GBA link cable uses a different voltage to the Gameboy link cable. Even though a GBA can play old Gameboy games, it uses different circuitry than when playing GBA games, so the games' link cable functionalities are fundamentally incompatible.
3. Even if these weren't an issue, Nintendo never released a product that allowed trade from one to the next.

To play old pokemon game you just need a Gameboy, any product labeled "Gameboy" that isn't the Gameboy Micro will suit your purposes. If you need a new one, your best options are the following:

Game Boy Color (not recommended)

There's no reason to get a gameboy pocket (or, barf, an original gameboy) since the GBC can easily emulate the monochrome display if you MUST not have color. Otherwise, the GBC allows gen 2 to be played in proper color as well as lending a less-impressive splash of color to the original games.

Also the GBC can easily trade via link cable to other gen 1 games, assuming you have friends to trade with (pretty unlikely these days).

Game Boy Advance SP (recommended)

Might be a bit more expensive, but the first built-in backlit and has rechargable batteries. Most people forget that old gameboys had disposable batteries and non-backlit screens. Only potential negatives are a possible cost increase compared to older models, and the GBA SP isn't quite as comfortable for extended use--I've only found this to be a problem with "twitch" games like Mega Man, shouldn't be an issue for Pokemon.

There are apparently "universal" link cables that let you attach a GBA to a GBC for trading like gen 1&2 require but I haven't used such hardware.

I know this isn't the question, but take heed that old gen pokemon games really don't hold up as well as you might hope. The PC box system was AWFUL in gen 1 and still poor in gen 2. I highly recommend going for the gen 1 and gen 2 remakes personally, they're available on GBA and DS respectively (FireRed and LeafGreen for gen 1), so a single DS system would be enough to play both of them, and all following generations up to 5.