Here is the general order of things before you reach the Forge to be able to forge your Weapons and Tools (I'll spoiler everything after the first):
1. Complete the Community Center (or complete the Joja Community Development Projects)
2. Fix Willy's Boat
3. Open up Ginger Island to where the North section is available
4. Make your way through all 10 Levels of the Volcano
You will find the Forge there. There will be a shortcut you need to open to get back there quickly without needing to do #4 every time. Don't forget to do this first!
For the Mini-Forge RennyBenny mentioned, you will get that pretty late in the game after you get all your game skills (Farming, Foraging, Mining, etc.) to Level 10. At that point, you will gain access to a spot in Cindersap Forest which will let you choose from some more game tools and options. After you gain even more XP of course... One of those options is the Mini-Forge you can use to forge your stuff in the comfort of your Farmhouse!