Should I be spreading out my skill growth at the start?

There's no real need to spread out or concentrate your skill growth. Fallen London is a game of exploration, not competition, and its main joy is in following stories, not raising your stats. Even death is merely a temporary setback.

At the very beginning of the game, your Training Profession raises its selected quality very quickly; it's pretty common to have one highway quality outstrip your other qualities early on. This one high quality is pretty well contained, though, and doesn't affect storylets that use your other qualities. The first several areas of the game each require only one highway quality: Watchful in Ladybones Road, Dangerous in Watchmaker's Hill, etc. Even as you move on to the second areas for a particular quality, you still need only that one quality: the Flit still only needs Shadowy, the Wolfstack Docks still only need Dangerous, etc. That will change slightly as you get to the third set of areas: the Labyrinth of Tigers is mostly Dangerous, but also has a few storylets that require Watchful or Persuasive; Mahogany Hall is split nearly-evenly between Shadowy and Persuasive.

The main thing you'll notice as you progress is that storylets pay out proportionally to the minimum quality required to attempt them. So the storylets your Watchful 30 lets you attempt might pay out 30 Whispered Secrets (or other rewards that give an equivalent of 30 pence: 30 Jade Fragments, 30 Rostygold, etc), while the storylets your Shadowy 5 allow will only pay about 5 Whispered Secrets.

It isn't a problem to have one quality much higher than the others: it just means that you will be able to follow that quality's stories much faster. If you are truly a min-maxer trying to get the best-possible profit as quickly as possible, then you'll raise one stat at a time via Training Professions, while always playing storylets from your highest quality. (You can leave your current Profession, and select a new Training Profession, at your Lodgings.) But if you do that, then you'll miss out on all of the wonderful writing of Fallen London: all of the early storylets will quietly disappear as your qualities raise higher and higher. You can temporarily make them reappear by equipping a Talkative Rattus Faber (purchased in the Bazaar tab in Nassos Zoologicals), which applies -25 to all of your main qualities.

My recommendation: play the Watchful storylets, before your Training Profession raises you above them, so that you can read the stories. This will still leave you plenty of time to play whatever other storylets interest you, regardless of what quality they require.