In D&D 5E, is there a way besides the Guild Artisan Background (PHB 132-133) to learn professional skills? i.e. blacksmithing, knowledge, gem cutting, etc.?
The short answer is "yes, in downtime."
Training in new tool proficiencies is explicitly covered in the Player's Handbook, pg 187, under "Training":
The training lasts for 250 days and costs 1 gp per day. After you spend the requisite amount of time and money, you learn the new language or gain proficiency with the new tool.
Dungeon Master's Guide
The Dungeon Master's Guide mentions training as a possible reward under the "Marks of Prestige" section (begins on page 228; training is discussed on page 231). No mechanics are given for this; it is entirely the DM's prerogative.
Xanathar's Guide to Everything
Xanathar's introduces more fleshed out rules for downtime activities, one set of which is provided under "Training" on page 134:
Receiving training in a language or tool typically takes at least ten workweeks, but this time is reduced by a number of workweeks equal to the character's Intelligence modifier (an Intelligence penalty doesn't increase the time needed). Training costs 25 gp per workweek.
Under these rules, the cost would be the same or less, and the time would be significantly less. However, Xanathar's also includes rules for downtime complications that can result; the examples given in the book are all related to something happening to or because of the teacher.
Practice, practice, practice....
You seem to be describing Downtime Activities, found on PHB p.187. One class of downtime activity is "Training," whereby you can gain proficiency with a tool, like blacksmith's or artisan's tools.
The examples given in "Training" are pretty thin. You'll definitely want to/need to work with your GM on this one.
As an aside, in one of my groups we rule that downtime accrues as time passes in real life. (In addition to as time passes in-game.) That way when it's hard to get synchronous games in, at least we're amassing currency for the asynchronous game =)
There are several ways to gain proficiency with artisan's tools.
- Through your race
- Dwarves automatically gain proficiency with Blacksmith's tools, Mason's tools, or Brewer's supplies.
- Rock Gnomes gain proficiency with Tinker's Tools.
- Through a background
- As you mentioned, the Guild Artisan Background gives proficiency with one set of artisan's tools.
- The Folk Hero Background also gives proficiency with a set of artisan's tools.
Through your class
- One of the starting proficiencies for Monks is one instrument or one set of artisan's tools. A Drunken Master Monk (see Xanathar's Guide to Everything) gains proficiency with Brewer's supplies from level 3. A Kensei Monk (also Xanathar's Guide to Everything) gains proficiency with Calligrapher's supplies or Painter's supplies from level 3.
- A Battlemaster fighter gains proficiency in one set of artisan's tools of their choice at level 3.
- From level 2, a Knowledge Domain Cleric can use their Channel Divinity to gain proficiency with any skill or tool for 10 minutes.
- A Forge Domain Cleric (Xanathar's Guide to Everything) gains proficiency with Blacksmith's tools from level 1.
Through a feat
- The Skilled feat grants proficiency with any 3 skills or tools of your choice.
- The Prodigy feat (see Xanathar's Guide to Everything) grants a Human, Half-Elf, or Half-Orc one skill proficiency, one tool proficiency, and fluency in one language.
- As other answers have mentioned, you can spend downtime training to gain proficiency with artisan's tools. The specifics will have to be worked out with your DM.