Where can I find the .apk file on my device, when I download any app and install?

There is an app in google play known as MyAppSharer. Open the app, search for the app that you have installed, check apk and select share. The app would take some time and build the apk. You can then close the app. The apk of the file is located in /sdcard/MyAppSharer

This does not require rooting your phone and works only for apps that are currently installed on your phone

All user installed apks are located in /data/app/, but you can only access this if you are rooted(afaik, you can try without root and if it doesn't work, rooting isn't hard. I suggest you search xda-developers for rooting instructions)

Use Root explorer or ES File Explorer to access /data/app/ (you have to keep going "up" until you reach the root directory /, kind of like C: in windows, before you can see the data directory(folder)). In ES file explorer you must also tick a checkbox in settings to allow going up to the root directory.

When you are in there you will see all your applications apks, though they might be named strangely. Just copy the wanted .apk and paste in the sd card, after that you can copy it to your computer and when you want to install it just open the .apk in a file manager (be sure to have install from unknown sources enabled in android settings). Even if you only want to send over bluetooth I would recommend copying it to the SD first.

PS Note that paid apps probably won't work being copied this way, since they usually check their licence online. PPS Installing an app this way may not link it with google play(you won't see it in my apps and it won't get updates).

You can use a file browser with an backup function, for example the ES File Explorer Long tap a item and select create backup