There are many site out there that have models that people have designed and shared. Many times even for what would seem to be a custom need that you have, someone has already designed it. Start off with these sites and then later you can get into designing your own creations.
www.thingiverse.com - ThingiVerse is one of the most popular site for models, but can be a bit disorganized compared to others.
https://www.myminifactory.com/ - My Mini Factory has designs that are guaranteed to print. Many are paid and you can upload your designs to make money off them. This is the gallery for the 3D design tool TinkerCad and Autodesk. This site also has design challenges with nice prizes if you are a designer.
https://cults3d.com/en - Cults 3D checks the models for printability and sorts them into categories.
https://pinshape.com/ - Pinshape has both free and paid models, so be careful which you fall in love with.
http://www.3dshook.com/ = 3DS Hook has more paid models. Also a place where you can start making money on your model if you get good at designing.
https://3dwarehouse.sketchup.com/ - 3D Warehouse is the gallery for Sketchup created designs.
https://www.cgtrader.com/ - CGTrader is a printable model database, but most come at a cost. Many jewelry items here.
http://www.yeggi.com/ - Yeggi is the google for 3D model databases. It searches all the others listed here.
https://grabcad.com/ - Nice engineering models, but not necessarily focused on printing. These you may have to modify in a CAD package before printing.
http://www.turbosquid.com/ - Turbo Squid. Last resort site. All models are paid and there is no way to sort by printable items.
In case I missed some, here are some more.
|Yeggi||Search Engine||Free, Paid||n/a||★★★|
|XYZprinting 3D Gallery||Marketplace||Free, Paid||★★★||★★★|
|Dremel Lesson Plans||Repository||Free||★★||★★★|
|STLFinder||Search Engine||Free, Paid||n/a||★★|
|3DShook||Subscription Service||Free, Paid||★★||★★|
|STLHive||ShopFree, Paid||Free, Paid||★||★|
|NIH 3D Print Exchange||Repository||Free||★★★★★||★★★★|