Do I need to buy approach shoes?
Approach shoes are useful in a very narrow cross section of climbing, and are not, in general, a tremendous help for the average climber or outdoors enthusiast.
This style of footwear is a compromise between a climbing shoe and a hiking boot, underperforming in each regard, and only valuable when you don't want to bring two pairs of shoes on your outing. They might be great, for example, on a long alpine ridge traverse with lots of hiking and climbing combined, or on a via ferrata.
You should consider the investment if (a) you're doing a lot of scrambling and climbing on 4th or low-5th class terrain, or (b) you're doing a lot of long, hard aid climbing where climbing shoes would be too uncomfortable to wear all day.
In recent years, the outdoor industry has used the technical applications of approach shoes as a marketing hook to promote the product as a general-purpose replacement for hiking or trail running shoes. There are now a gazillion types of approach shoes to choose from, with several new models looking exactly like hiking boots with an extra label attached.
For recommendations on what approach shoes to buy, we like this write up by Switchback Travel: http://www.switchbacktravel.com/best-approach-shoes
As well as this (somewhat dated) survey by Outdoor Gear Lab: http://www.outdoorgearlab.com/topics/climbing/best-approach-shoes