KSP 1.4 and the DLC come out later this month. In preparation for that, I present my recommended mods for 1.3.1:
I've played a number of different KSP mods, and I'd like to recommend a few:
BlueDog Design Bureau
BDB is based around providing the parts needed to fly pre-shuttle US space launchers, and some of the 1960s probes. Included is Mercury/Redstone/Atlas, Gemini/MOL/Titan, Apollo/Skylab/Saturn, and many others. BDB, like many of the other mods on this list, are done in what is called a 'Porkalike' style, so named after the KSP developer Porkjet, who came up with a cohesive style for all of the spaceplane parts that are in game.
A subsidiary of BDB, focuses on providing a usable X-20 that integrates with BDB. Currently in development, but flyable (needs landing gear and adapters).
Surface Experiments Pack
Ground science experiments to make it worthwhile to visit bodies of the Kerbol system. Requires KIS
Like SpaceX? Tundra adds SpaceX
What BDB is to the American Space Program, Tantares is the to the Soviet one. Full of parts needed to fly Soyuz, Zond, Salyut, & Mir missions, and with R-7, N-1 & Proton rockets, Tantares is very good for your inner Soviet Cosmonaut.
When the developer of Tantares dropped Vostok from his mod, another user picked it up and ran with it.
Tantares in Colors
Stock Tantares is rather grey. This opens up a wider color pallet.
Shuttles! CA adds Kerbal-alike versions of the US space shuttles with both SSRMs and LRBs. Also included are parts for the Block II orbiter, and Buran (without Energia at this time). Fits with BDB's Centaur G' nicely. Soon to have Soviet LKS and US IUS.
Coatl Probes Plus!
Offering the probes that BDB and Tantares don't (based around historic craft), Probes plus gives you options for filling out your uncrewed program.
Commonwealth Rockets/British Rocket EXpansion and Integration Technologies
BREXIT gives the player the few British rockets that were built and were proposed. By the same people that made BDB and Tantares.
While you can build the Russian Orbital Segment of the ISS with Tantares, HabTech lets you build the US Orbital Segment. Comes with Trusses, Solar arrays, and station modules. Very low memory relative to other station mods.
At this point, we get away from historical rockets, and start getting in to near-future items:
Stockalike Spacestation Parts Redux
Offering a bridge between stock and Nertea's Near Future and related mods
, SSPXr includes inflatables, large (3.75m) and small (1.25m) station parts. I find myself using SSPRx to give some pizazz to my Freedom-esque stations.
Kerbal Planetary Base Systems
At some point you want to do more than just fly around in space, and this is where KPBS comes in. It offers the player the ability to design, fly and land modular outposts. Excellent textures and IVA views.
Feline Utility Rovers
By the same developer as KPBS, FUR adds larger rovers for far reaching exploration and science.
Grounded! Modular Vehicles
Around KSC there are a number of vehicles (all date back many years to the work of Bac9, the guy who designed all of the late game buildings) as static objects. Sometimes however you may feel the need for a ground race, and Grounded offers that, without the size of FUR. Note that FUR and Grounded! play very nicely with each other, and sort-of work, even with two different design aesthetics.
Racing is all well and good, but sometimes you want to fly under the bridge in the science center, or just have more options for flying around Kerbin and Laythe. This gives you that, while looking good to boot.
Kerbal Joint Reinforcement
While some may find the spaghetti rockets fun, many don't, and of those, many don't find having to auto-strut all of the parts together anything but annoying. KJR does that for you.
KSP Trajectory Optimization Tool
Ever wanted to fly an Eve-Kerbin-Kerbin-Jool set of Gravity assists? KSPTOT is the mod for you. Uses the Matlab runtime to generate flight trajectories that can be imported back into the game.
Kerbal Engineer Redux
or Dmagic's Basic mods
or Mech Jeb
KSP doesn't include features such as a DeltaV calculator on the fly, so either of these mods will do it for you. MechJeb also has advanced integrated auto-pilot capabilities.
Kerbals are about .62-.64 times the size of humans (linearly). When you apply that measure to Kerbal rockets, they are over-scaled relative to their system (ie a Saturn V needs no third stage to throw a CSM/LM to the moon). Kopernicus is the base mod for changing the Kerbol system, and lets you scale the system up. I use a 2.5x scale most of the time. While it makes the game harder, it also adds a slightly greater sense of accomplishment.
KSP's rocket exhaust effects are bad. Real Plume makes them better.