I'm a paint fan - I hope I can do this clearly.
Paints for styrene (or anything really) have pigments (what colors them) and an agent (what the pigment is in). Both agents and pigments vary - but what really counts usually is the agent.
Enamel paints have a chemical solvent as an agent. They are the oldest type of paint used on styrene. They render good color and can be hand or airbrush painted well. They take a long time to dry, have a very strong odor and are tough on clean-up. Many of us started with enamels years ago, (hand painted natch - when I was 12 I'd never heard of an airbrush) but few use them anymore. There are many enamel based weathering products that are effective if you don't mind the smell and mess. Testors, Humbrol and Colourcoats are the only enamel paints I know of still sold.
"Acrylic" paints are a real moving target and vary greatly in agent and pigment. Acrylics dry much faster than enamels, have a more muted odor, and, at least to some degree can be cleaned up with water. The most popular acrylics are solvent based - this means Tamiya, Gunze and a very popular new paint from Eastern Europe MRP. Because they're solvent based they do have an odor, but it's not too bad. (You can tell a solvent based paint because on its label it will say "flammable" and "toxic.") You thin solvent based acrylics with solvent thinners - many people use the thinner made by the company. However, I'd use "lacquer" thinner - both Gunze and Tamiya make it. I think it's much better than Tamiya's "acrylic" thinner A-20. I think that if you don't mind the slight odor that the solvent based acrylics have they are the best paints available for the airbrush. You'll have to decide on your own ratio of paint to thinner for good results - the type of airbrush has impact there. But these are very good paints for most purposes - Tamiya is easily found is made by the best modeling company on earth. None of the solvent based acrylics are very good for brush painting - some of the qualities that make them good for airbrush actually work against them. (Should note that the company AK is now making a line called "Real Color" which are very like Gunze and are widely admired for the accuracy of their military colors. Don't know how important that is for WWI topics.) You can hand brush with solvent acrylics, but they don't excel.
Water based acrylics have an agent that is a kind of watered down plastic polymer. (Water based acrylics are not flammable nor are they toxic - hence their popularity in schools. Some of their pigments, however, you wouldn't want to inhale, so in theory, you should use a mask while airbrushing. I don't.) If you could see it, it's white. These are very different paints and require different techniques. You can thin water based acrylics with water - but only a little bit - maybe 20% at most. The most prominent water based acrylics are made by the Spanish company Vallejo. They are sold as Model Color and Model Air. Model Color paints are pretty thick and really made for brush painting at which they excel. Model Air has a greatly thinned polymer and is made for the airbrush. Vallejo makes a thinner which I'd certainly use - but use it instead of water - only 20% max. (Take my word for it: if a polymer based acrylic is overly thinned with water the finish will be very fragile and may not really take at all.) A better thinner I think is what Vallejo calls "Thinner Medium" which is white - and is really a very thin version of the polymer agent. Art House brands like Golden and Liquitex make better "Airbrush Medium" (same white stuff, just with better ingredients) which is what I use. (Golden is to acrylic paints what Tamiya is to models. Golden has a bewildering variety of acrylic paints and mediums - they aren't cheap but they are the best. Mind you, artist materials are all cheaper than modeling materials because they sell it in much larger quantities. If you log onto the big Vallejo web site, your first choice is to choose between "Arts" and "Models." I wish Golden knew how much paint they could sell to modelers - or maybe we really are a small market.) There are other water based acrylics: the Italian company LifeColor has a huge number of colors and a loyal fan base, although they have a learning curve with the airbrush. The American company Mission Models makes very good water based acrylics too, but you must buy their thinner and resin to go with them. There are many colors. Ak Interractive also makes water based acrylics - their team was formerly from Vallejo. The best model paints are from Golden - but you have to dispense with military colors, which most modelers won't do. I like to mix my own colors, so they're great for me. Golden High Flow are terrific in an airbrush. All water based acrylics hand brush very well. You will want some of these as hand painting will always come up. People who do figures will have a huge array of paints like Model Color (and, if they're smart, Golden Fluid Acrylics).
Lastly there are lacquers. They have a solvent agent that's "hot" and pretty much demands a model company's thinner. (Never use a hardware store lacquer thinner on plastic - it will scar it badly.) Because they're "hot" they actually bond directly to the plastic itself. Lacquer paints - Alclad is the most famous - are good for metallics but have a very powerful odor. There are people that argue that Tamiya and Gunze as at least "half lacquers" - but I wouldn't worry. I use a automotive lacquer primer which I like very much - but only outside.
Oils - Increasingly oils paints are making their way into the modeling world. If you use them at all - this would be for weathering or trying to replicate wood grain - you don't need many colors. You also don't need big tubes. For most purposes "student" grade oils in small tubes are fine - you will need odorless "White Spirit." I use Gamblin "Fast Matte" oils paints (which dry very quickly) and employ a terrific spirit Gamblin calls Gamsol. Not cheap, but the most benign spirit on earth.
You can't go wrong with Tamiya - unless you're picky and want to master water based paints. But what colors? Model companies suggest colors and usually reference them to a particular type of paint. Wingnut Wings gives their colors in Tamiya and Humbrol. Eduard gives theirs in Gunze. There are web sites that have ways to figure out what a Tamiya paint is in another color: (For instance Tamiya XF1 Flat Black is Gunze C83.) I might add that with Tamiya you can buy smaller bottles - in general I'd recommend them, unless you're dealing with very common paints like black or neutral gray. Paints don't improve with age, especially when opened.
Best of luck.
Eric