R&D Agent Basics:

Train Science 5.
Raise standing with a corp that has R&D section.
Find an agent and train at least 1 lvl in a shared field/skill.
Then sign on.

Listed below is the State, the corp, and the major fields of research with those corps:

Ammar Empire

Carthum Conglomerate - Laser Physics & Amarrian Starship Engineering

Viziam - Amarrian Starship Engineering & Nanite Engineering

Caldari State

     Ishukone - Hydromagnetic Physics & Caldari Starship Engineering

     Kaalakiota - Electromagnetic Physics, Graviton Physics, & Caldari Starship Engineering

     Lai Dai - Caldari Starship Engineering, Graviton Physics & Rocket Science

Gallente Federation

     CreoDron - Electronic Engineering, Caldari Starship Engineering & Gallente Starship Engineering

     Duvolle Laboratories - Plasma Physics & Gallente Starship Engineering

     Roden Shipyards - Gallente Starship Engineering, Plasma Physics & Quantum Physics

Minmatar Republic

Boundless Creations - Mechanical Engineering & Minmatar Starship Engineering

Core Complexion - Minmatar Starship Engineering & Nuclear Physics

Others

     Khanid Innovation (Khanid Kingdom) - High Energy Physics & Amarrian Starship Engineering

     Thukker Mix (Thukker Tribe) - Molecular Engineering & Minmatar Starship Engineering

     Nefantar Miner Association (Ammatar Mandate) - Laser Physics, Amarrian Starship Engineering, Molecular Engineering