Unfortunately, Puma is more of a fashion brand and not particularly respected when it comes to running shoes.
I have to buy 3E-4E width (H for those of you in the UK) shoes so I understand the struggle of finding shoes that fit.
As far as what kinds of shoes are best for walking and running? My answer would be different depending on if we were talking about walking, short distance running, or long distance running (10km+). I own about 15 pairs of running shoes and each with different purposes.
First and foremost, get a shoe that fits. Men's shoes are wider than women's. They do make wide width women's shoes, but the selection is often limited. I buy my running shoes a size larger because I don't want my toes hitting the tip of my shoe. That leads to blisters and leg pain over time.
Second, don't buy cheap quality shoes. A shoe that retails for 50gbp is probably much lower quality than a shoe that retails for 100gbp but is marked down because it's an older model. I always buy older models because the quality is unchanged (only the cosmetics), but the price is usually 50%+ cheaper.
Third, good support and the weight of the shoe are issues if you plan to run longer distance. For example, my Mizuno Wave Inspires are great up until about 6-8 miles and then they start to weigh me down.
Last, a lot of it is personal preference. Many people buy different brands and models of shoes until they find the one they like. After that you just need to replace them when they wear out,