Which size to buy? Most adults are going to have a harder time playing the smaller soprano (20") size. The main reason they are popular is because they are often the least expensive of the three main sizes.
If you've played guitar in the past, you'll most likely prefer the larger tenor (26") size. If you've never played an instrument before and your hands are not too large, then you will probably prefer a concert (23/24") size. Avoid the baritone ukulele because it's actually tuned like a guitar, not a ukulele. Baritones are for people who play guitar and like the sound of a ukulele, but want to stick with chords they learned on the guitar. None of what I teach applies to baritone sized ukuleles.
If you need a slightly larger size, the Oscar Schmidt ukuleles are specifically designed for larger fingers.
Two things I look for in a quality ukulele:
Chrome tuning pegs (not plastic)
Matte finish (not laminated)
Produces a deeper louder sound from the wood used. Lamination tends to dampen the sound.
Also - a good digital tuner is the Snark. You can usually find one on Amazon for $15 or less. It's ok to use the guitar tuner version of Snark. It'll work fine for your ukulele.
Affordable starter ukuleles
You can easily spend $500 or more for a ukulele. But if you just started playing, I am not sure why you'd spend so much so soon. Invest in some lessons and songbooks instead! Then when you get good, a few years down the road, you can upgrade to something nicer. Prices for a high quality ukuleles are $75 - $150. Anything over $200 is not what I would consider a 'budget' ukulele. Reputable brands include Alverez, Oscar Schmidt, and Kala.
Kala recently released a Bamboo Uke!
Q. Do I need a strap?
You might consider getting a strap. Nothing worse than dropping your ukulele! Even if your ukulele doesn't have a strap button (most don't), it's easy to install one. I personally don't use one when I teach because I have to put down my ukulele so often to write things down or use my computer.