Hi,
I remember the grubby socks well with my DC! We decided to go for barefoot shoes because up until children are about 5 or 6, their feet really shouldn't be in supportive shoes.
This is a newspaper article but it does have a few references on it, and there has been lots of research recently.
For first shoes, my girls both had the little soft leather type that babies often wear, as they keep their feet clean, and allow them to develop naturally. I mean this sort of thing. They had outdoor pairs and indoor pairs. They are not waterproof, but at that age they didn't walk in the wet much. They later had wellies for puddle jumpin (not strictly speaking soft soled, but they only wore them for short periods of time).
When they were a little bigger, say, walking to the car to get in, we moved on to Little Blue Lamb shoes, which you can buy through ebay, or I have found two websites that sell them "Kidz Shooz" and "Cheeky Rascals". Little Blue Lamb make different styles as your children develop in their walking confidence. We have always been very happy with them, especially the price!
The other place that is really good is Happy Little soles which stock more high-end barefoot shoes, with some absolutely gorgeous ones from Livie And Luca, Bobux etc.
Even before I had children, I always thought that Clarks and StartRite were uninspired shoes! Pink or navy or black, basically! And the price is astronomical!
For measuring, you can either buy a foot measure online, or the websites explain how to measure without. I sometimes cheekily go into clarks and get them measured, then buy online 
. We have only once had a problem with the fit, and the website accepted and exchanged the shoes for the different size with no quibble and far easier than lugging a toddler into town!
Wow, what a long post! It's something I am quite passionate about, though, so hopefully it has been interesting! 