The best first step into woodworking as a career is to complete at least two years, ideally three, of City & Guilds joinery or furniture making. Although these courses are listed as full time they are generally two days a week, which opens them up for day release or for people earning some income on a part time basis. As long as you're hale and hearty age isn't a barrier.
There are also ever increasing numbers of workshops offering paid for tuition. Some, like Peter Sefton or Waters & Acland are as diligent and professional as you could hope for, providing a well thought through structured training programme. Others, including unfortunately some fairly big names and regular advertisers in the woodworking press, are not much better than disgraceful. Bad practises abound, including packing in too many students with too little personal tuition, a slapdash approach to safety, proprietors who spend all day surfing the web while encouraging excessively large/low skill projects that are basically designed to keep the students occupied for many hours without much direct tuition. Sadly, I could go on, I've met several ex students from some of the worst training workshops who have wasted a great deal of time and money. The saddest are those who don't even know enough to know they've been had!
In the middle are a number of training workshops that are reasonable, but have some significant negatives you'll need to think hard about. What would happen if it's a one man band and the proprietor fell sick? Do they give relevant training or are they skewed towards something like antique restoration that doesn't particularly interest you?
The other thing you need to be realistic about is just how equipped you will be after just nine or ten months training (the typical "long" course). In a decent school you'll have covered the very basics, but you'll still be desperately slow and with some huge gaps in your knowledge. The traditional seven year apprenticeships might have included a lot of time "on the broom", but you'll still need approaching 10,000 hours of experience under your belt, about five years, before you can consider yourself a craftsman able to command full wages.
Finally it's worth saying a word or two about the jobs market. There are lots of wood related careers. You can earn a decent wage as a kitchen fitter (if you don't mind dealing with the other trades and manhandling Aga cookers around), similarly you can support a family as a shop fitter (providing you're okay working away and living in digs), or you could be a wood machinist or a install yacht interiors or a million other options.
But when most people talk about a career in woodworking what they really mean is working on their own, designing and making solid wood custom furniture in a converted barn surrounded by beautiful countryside. Sadly, the designer/maker dream is just that...a dream. Unless you have a partner with a real job, or you've taken early retirement from the forces or police and have a basic living pension, then the reality of making hand made furniture for a living is that in a good year you might earn the minimum wage, and in a bad year you'll go bankrupt.