First off, I'm no expert on compressor/air type sprayers. Most of the time, these type sprayers are used for very thin paints such as automotive, lacquers etc. They also have fairly small paint reservoirs attached to the gun itself. They tend to put out a lot of over spray and should be used in a controlled area like a spray booth with proper ventilation. I never see this type of sprayer on building jobs.
The airless sprayer, such as the one you noted, use a high pressure pump to push paint through the hose to the gun. They deliver a good pattern (depending on tip selected), are easily controlled and since they don't mix air with the product, have far less atomazation of paint in the air. the other huge advantage is that the pick-up hose can be put directly into your 1 or 5 gallon can of paint. Follow the directions on priming and cleaning, but it is easy. I personally have a Graco model 17 and love it. It is a professional grade sprayer and retails around $450-$500. Just two weeks ago, I and a helper painted and back-rolled a complete two story house and detached garage in one day.(very long day, however!) This saved 3 or 4 extra days of labor and the results were fantastic. We shot both semi-trans stain on the house and matching color solid stain on the garage. This sprayer handled both products easily. (I always use apx 10 to 15% flowtrol with thicker solid latex based products)
The only disadvantage of this type of sprayer is on very small jobs. It takes about a pint or more of paint to fill a 25 foot hose and closer to 1/2 gallon for a 50 ft hose when priming the unit. This amount product is usually wasted at the end of the job when you get ready to clean the pump.
From the type of jobs you plan to do, I would recommend the airless sprayer hands down.
There are (at least) two different types of stains for concrete. One is a surface coating, the other is acid etching.
The former can work, but isn't as durable/lasting. I believe most of the products you get at the big-box stores are more of a top-coat type of stain.
Acid etching is just that, it's a stain containing an acid that actually etches the concrete. This is the typical type of stain you see in home magazines and such. The etching provides a deeper stain that is more durable and not prone to flaking.
I've done the latter, and applying it is super-easy. It's the prep work that can be a chore--namely if you have adhesive on the concrete already. In our case, we has tile mastic, which we had to dissolve using soybean based stripper. It worked great but was incredibly messy.
If you don't have glue to deal with, however, you should be good to go. Order up the stain (lots of places online to get it). You then dilute it per instructions and spray it on the concrete. I used a garden sprayer for this. You then let it sit for a period of time, and then you apply a neutralizer to stop the reaction.
After that, you apply water and mop/squeegee/wet-vac it up.
Then, you apply a sealer of some sorts. Common options are wax-based, polyurethane, and epoxy. I went with poly as it was relatively easy to apply (roll it on). That said, I don't think it was ever truly waterproof. For your needs, you may want to go the epoxy route.
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It depends on the type of wood and existing treatment on it. If it already has paint on it and you are painting a similar color it won't need as many coats as fresh wood.
You have about 3100 SF to paint and you don't have any edging to worry about and it sounds like the everything is accessible. When you are painting outside you can be a little messier than inside a house so its not unreasonable to get 400-600SF painted in an hour when hand painting. Which would get you about one coat a day on the fence.
Paint sprayers are great if you are doing a big area, but outside can be a burden with wind, bugs and pollen. They also typically put the paint on thinner than if you roll or brush it on. Or you can end up spraying to much on and needing to smooth it out with a roller.
Depending on what sprayer you get they can also require a fair amount of clean up and setup which is something to consider. The first time I used a sprayer I covered myself in paint, made a mess, and generally created way more work for myself than needed. After that I was way more cautious about completely prepping areas before hand, getting the right spray consistency and not rushing myself with the sprayer.
I've found the sprayer is best for interior work laying down primer or white base coats before flooring or a lot of other finishes are in.