The first step in changing your profession is leaving your current one.
You can make a resignation letter for yourself by following the "Choose a new Profession" Storylet located at your Lodgings (note that if you are resigning from an Advanced Profession, you will lose the unique item associated with it.):
Once that is out of the way, you can then take up a new Training Profession or Advanced Profession.
Training Professions help improve your main qualities up to level 70. After that, you'll want something a little more advanced. It's a little out of the scope of this question to list the details of every possible profession, and there is already a good article about it that you can go check out at the Echo Bazaar wiki.
To gain a new Training Profession, just use the "Adopt a Training Profession" Storylet that you can find at your Lodgings after quitting your old Profession.
Advanced Professions require you to have connections with certain factions, and opportunities to gain a new Advanced profession are only available through "Connected" type Opportunity cards.
If you don't want to wait for a certain Opportunity card to show up, you can spend some Fate on the "Arrange Favourable Circumstances" Storylet and use the resulting Favourable Circumstances to arrange a meeting with your contacts using the aptly named "Arrange a meeting with your contacts" Storylet.
Both of these storylets can be found at your Lodgings, under "Write Letters":