How to change your profession skills

Upon reaching level 5 and level 10 in each profession, you get to pick between two different skills. The level 5 skill will affect which skills you can pick up at level 10, and a lot of people wants so change their skills.
It is currently no in-game way to change them, but ConcernedApe says he is interested in making that available in the future.

Not being able to do it in-game does however not mean it's possible, and at the moment you can change your skills by editing your save file! This means editing the files where your save is located, so you should make a backup before you start editing, so that you can revert the changes in case you mess up.

Changing your profession skills

The first thing you need to do is find the folder with the save data. For Windows users its usually something along the lines of C:\users\Your_Windows_Username\appdata/roaming/StardewValley/saves/CharacterName_Number/

 Upon up the file called charactername_number with Notepad or another text editor and scroll down to the part that says <professions><int>7</int></professions>. Change the number 7 or whatever number you already have to match the profession skill you want using the table below.

Farming:
0 = Rancher: Animal Products worth 10% more.
1 = Tiller: Crops worth 10% more.
2 = Coopmaster: Befriend Coop animals quicker.
3 = Shepherd: Befriend barn animals quicker.
4 = Artisan: Artisan Goods(wine, cheese, oil, etc.) worth 50% more.
5 = Agriculturist: All crops grow 10% faster

Fishing:
6 = Fisher: Fish worth 25% more.
7 = Trapper: Resources required to craft crab pots reduced.
8 = Angler: Fish worth 50% more.
9 = Pirate: Chance to find treasure doubled.
10 = Mariner: Crab pots never catch trash.
11 = Luremaster: Crab pots no longer need to be baited.

Foraging:
12 = Forester: Wood worth 50% more.
13 = Gatherer: Chance for double harvest of foraged items.
14 = Lumberjack: Normal trees occasionally drop hardwood.
15 = Tapper: Syrup is worth 25% more.
16 = Botanist: Foraged items are always gold quality.
17 = Tracker: Locations of foragable items are revealed.

Mining:
18 = Miner: +1 ore per vein.
19 = Geologist: Chance for Gems to appear in pairs.
20 = Blacksmith: Metal bars worth 25% more.
21 = Prospector: Coal find doubled.
22 = Excavator: Geode find doubled.
23 = Gemologist: Gems worth 20% more.

Combat:
24 = Fighter: All attacks deal 10% more damage. +15 HP
25 = Scout: Critical strike chance increased by 50%.
26 = Brute: Damage is increased by 15%.
27 = Defender: HP is increased by 25.
28 = Acrobat: Cooldown on special moves is cut in half.
29 = Desperado: Critical hits are deadly.

Now just save the text file and restart Stardew Valley, and you have your new profession skills ready!

Source:  /u/Froghat_Vendor at Reddit.