Regarding using Kindle for PDF: Kindle doesn't let you search or highlight words on PDF files. Kindle transforms every page of a PDF file into static screen. Better use the official Adobe Reader app to open PDF files, where you can search, copy, or highlight words.
Regarding putting a PDF file shortcut on the home screen, you can just install the Adobe Reader app and put it on the home screen. Opening Adobe Reader will show you recently opened files, or all PDF files found on your device including SD card.
To answer the OP's question, after you install Adobe Reader (or any PDF viewer app), you should be able to download PDF files from the web. You should be prompted with the request to open the PDF file with your viewer app. Opening it will also save it to your device's "download" folder.