Compared to writing with a ballpoint, you have to exert almost no downward pressure when using a fountain pen. Indeed, many pens will write with only the pressure that the mass of the pen itself creates. This dramatically reduces hand strain.
I have quite a few fountain pens; there isn't a one that I would worry about keeping in my shirt pocket from the standpoint of ink leaking. (Air travel is a different issue; I've never had a pen leak on a plane, but I am more cautious anyway.) I've never seen a fountain pen that had a clip where the clip wasn't oriented to stay tip up when the pen is clipped.
These days, the standard first pen suggestions are either the Lamy Safari that others suggested or the cheaper Pilot Metropolitan. (The Metropolitan is available in North America; the Pilot MR is a very similar pen available in the UK and Europe, I believe, differing in using international cartridges and converters, rather than Pilot's proprietary ones used on the Metropolitan.)
Serious brick and mortar pen stores are a dying thing. If you have one near you, you are lucky. Amazon, Jet Pens, and Goulet Pens are the three online vendors where I've done most of my pen shopping. Jet Pens focuses on things Japanese; Goulet Pens has focused on high levels of service directed at those new to fountain pens.