One of the things I realized after buying lots of tools is that, most of these tools you can find replacement in your kitchen. I would highly encourage trying out different alternatives before spending big money on a fancy tool. The best tool is always your hands.
Here are a select few of the tools I use most often in throwing and hand building in the studio. If you have any suggestions or die-hard must-tries, please let me know in the comments!
Brush: from Michael's, used to apply slips to adhere pieces of clay together. You can get the cheapest brush you can find, and it will work just fine :)
Fiskars Exacto Blade: for hand building & cutting clay slabs. These are very sharp so it helps not distort the clay.
Spray bottle: from Grove Collective, a friend left it to me when she moved to Italy. I really like this bottle because it's made of glass, and the sprayer head works really well (unlike the cheap ones from Walmart). But the ones from the grocery stores work just as well.
Circular Clay Hole Puncher: I have a variety of sizes from amazon. This is the most frequently used size to cut out drainage holes for planters. But if I were to buy them again, I would buy this one which is half closed so the clay does not stick to the inside of the cutter.
Wood modeling tool: I got from local pottery studio. One of my favorite tools to use. I use this to trim when throwing, and use the smaller end when sculpturing.
Ruler: to measure (and imagine) clay shrinkage.
Needle tool: my most used tool. It's versatile. I use it to trim, scratch/scorch the clay to attach pieces. I also use the dull side to make indents for sculpture.
Mudtools Yellow rib: to scrape excess slip off the mug when I am on the wheel throwing.
Sponge: to clean the pieces and EVERYTHING else. You can buy them fairly cheaply from hardware stores (Home Depot & Lowes for the U.S.)
Mudtools trim tool: used for trimming on wheels.
Wire cutter from Mudtools: to cut clay, but I think any wire cutter would do. I've seen some made with fish line tied to two pieces of wooden dowels.