Take him to a shoe shop which sells clarks shoes. They're good because they do through fittings and sell half sizes in a variety of widths. They also have shoes for babies who are cruising which are softer and more flexible, so just what your DS needs right now if he's out of the house.
Round here, the 'first shoes' range are about £20, so a little cheaper than those for slightly older children, and a lot of places offer a presentation box to keep them in later, so it's not a bad price all included (though once you start buying them regularly it does add up).
If you're only putting them on him so he can wander around in the park, it might be worth getting him measured in clarks and then (assuming he's a 'whole' size) buying cheaper shoes elsewhere, but that's really dependent on your budget. Personally I try and buy 'good' shoes, but then our DS was a bit later starting to walk but fast learning to, so once he was on his feet he was keen to be on them a lot, and we felt it was justified.
The main thing to remember is that cruising shoes are much bendier, and usually not as big and 'clompy' so they aren't too heavy for their feet to lift. Expect a spaceman walk at first while he adjusts to what re effectively pretty big weights on the end of his legs all of a sudden, and don't buy them if you think it will put him off walking. He has to be ready to learn to walk in shoes, as it is a slightly different skill to walking barefoot. But the clarks staff are generally good - the woman we had said although our DS was walking when we got his, he was better in cruising shoes as they weren't as heavy (and they were cheaper, so that was fine by us!). They sell crawling shoes too, which I assume are lighter again, but we didn't really look at those.