To my knowledge a home office supply store like Home Depot or Kinkos will not have the paper weight appropriate for printing invites. You want a 100 lb or a 130 lb Double Thick Cover paper ideally. If you are going with a professional printer, then they will be able to show you paper samples and provide you with the right paper. Most paper companies don’t deal retail at all unless you look online. Strathmore is a good brand for invitation paper as is Classic Crest. Also yes you are right- the heavier the paper, the thicker it is, so it will not run through your home printer but since you are going to a professional printer, this should not be an issue. If you still want to buy them online, check out www.minted.com - they have lots of invitation options and they’re not super pricey, but again, it depends on your budget.
Also for flat printing- you have two options- you can print digitally which means it prints on a specific weight of paper, like a postcard, and can be glossy or matte in finish. You can use as many colours in a digital design as you like and your files can be created in Photoshop or Illustrator and sent to the printer as a high quality jpeg. Digital printing is basically accomplished with a glorified colour copier and its ridiculously cheap but it may not be suitable for a formal occasion like a wedding.
Offset printing is your other option- for this you would have to find a commercial printer. Offset costs more but you get a better result as you can also choose your paper. Offset is usually limited to 1, 2, 3 or 4-colour printing. The fewer colours you use, the less the cost. This is where you would be able to use 100 lb, 130 lb Double Thick, or even thicker paper if you so wished it. Keep in mind, the heavier the paper, the higher the cost. In offset printing, you can also use metallic ink which is a nice touch- its not super glossy but more subtle and you’d need to get your hands on a Pantone Metallic Swatch Book to see your colour options. Offset designs need to be created in Illustrator or InDesign and output as high quality print pdf’s using vector graphics.
Where are you located? If you’re in Toronto, I can likely show you all these things from what I have and recommend a printer to you.