Good, modest priced, brands readily available over here (UK) include Kala, Ohana and Mainland. There are some really great budget instruments coming out of China these days.
Ohana and Mainland are very similar. I believe they are actually made in the same factory. Quality control on both is excellent. Main differences are minor cosmetic variations - binding, tuners etc. The actual body patterns are much the same. Kala are a little different, but still similar.
A solid mahogany model from any of those would be a very fine starting point. I've played them all and they are all easily up to gigging standard and, IMHO, represent probably the best value for real, solid wood ukes. My personal favourites are the Kalas - I like the necks.
Be aware: a new uke will likely play dreadfully out of tune until the action is properly set at the nut and bridge. The combination of short scale and nylon strings is not very forgiving of high string action. Stock strings are nearly always crap. A decent set of Aquila or Worth strings makes all the difference.
I'm a moderator over at the Ukulele Underground forum, and I totally love 'ukulele. It's not my main instrument, but it is fun, versatile and addictive.
Google "southern ukulele store". Or, for more choice, try one of the German sites such as ukulele.de.