I just picked up an Android 2.3 phone - an HTC Evo for $30 on Craigslist - I'm sure similar phones are available in your city. It doesn't have phone service but everything else works great. I installed Tasker, Google Drive, My Tracks, Daily Roads Voyager, and Ulysse and just for fun Google Music and Zplayer. I also installed Launcher 8, a Windows phone type launcher because of the nice clear big buttons and it allows me to create a Car Dock style home screen, and the Tasker plug in for My Tracks which lets you automate stopping and starting the recording in the My Tracks app.
My Tracks uses your phone's GPS to record your position, speed, and heading extremely accurately; it was designed for runners but is well suited to automobile speeds. It can display your trip info as a chart or as a moving overlay over a Google Earth type image.
Daily Roads Voyager (DRV) uses your phone camera to video your road trip. You can select low or high resolution but for just recording any possible accidents and incidents low res is fine. It starts a new video by default every five minutes, can be set to retain the last several files or you can touch the screen at any time, after a road incident for example, to permanently save that video. Sudden deceleration such as braking prior to an accident causes it to retain the video. It's also very configurable with auto start and stop on power connection (eg starting your car).
Ulysse is a GPS speedometer. The paid version minimizes its entire display except a dime-sized speedo which overlays over any app you tell it to.
Tasker, Drive, and Google music you already know.
Zplayer is a music/podcast player which is based on the old Zune player.
I mounted the Evo by the driving mirror with the camera pointing forward using one of these sucker type phone holders from Walmart. I set a Tasker profile "Driving" to activate when I start my car. When the phone gets power the Driving profile turns on GPS (Android 2.3 so no user confirmation required), Bluetooth - for streaming music, Ulysse, My Tracks and DRV so from the get-go the phone is recording my speed and location and videoing my trip. A data connection isn't needed, not even for My Tracks which just records your location at short intervals so a 'phoneless' phone is ideal. The My Tracks data can be paired with a map when you have a data connection, in fact it happens automatically. Ulysse isn't actually needed as part of this setup- DRV has an on-screen, if smaller speed display but I had already bought Ulysse previously to this so I included it.
Instead of having Bluetooth come on as part of the Driving profile - I use it for Navigation - you could have it come on when you start the Navigation app or a Music player (see below)
My real phone has a Tasker profile that turns on the mobile hotspot if I plug it into my car charger. I have a Tasker profile on the Evo which detects when my real phone's mobile hotspot is near and switches on the wifi on the Evo in case I want to stream stuff from Google Music over the Bluetooth. When I stop the car and the power quits to the device a second Tasker profile executes which switches everything off. DRV has the option to turn itself off at power off and save the current recording so I go with that as Tasker just kills the app without saving the last video. Once the phone detects my home wifi it turns on wifi and will (I actually haven't set this up yet) upload videos to Youtube and save My Tracks to Drive. The Evo's single processor coupled with Android 2.3 has no problem running all these apps concurrently, as well as running Google Navigation and streaming music - this was a pleasant surprise, I expected a bit of lag. For those so inclined it makes a good way to record your road trips, for your own amusement, indemnity or if your employer has placed a speed recording device of some sort in your company vehicle. The videos can be placed in Google Drive or uploaded to your YouTube account. A good use for your old Android phone or one you can get for cheapies.
And there you have it; a device that records where you've been, tells you where to go (not that I need that) and plays a soundtrack all the while.
These apps are free - My Tracks, Drive, DRV, Google Music, Launcher 8.
Zplayer - at the moment is $3.44 but not needed for the Dashcam part.
Tasker you probably have but is $2.99, the My Tracks plug in is $1.49.
Launcher 8 has a free version with ads which works perfectly well in this scenario.
Probably the only thing you'll have to pay for to set this up is the Tasker My Tracks plug-in.