If I disable Location History and keep Location Reporting enabled, will Google still save the Locations I've been to? I am purchasing an Android phone for the first time and I am quite paranoid about security.
If I use plain GPS and not Google Location Services, will I be missing anything?
Is using Google Maps on Android secure, or should I use any alternate apps. I am looking for an offline GPS that I can use for apps like Runtastic etc.
Summing up all the comments:
Will Google save the locations?
Definitely. As soon as you want to use the more battery-friendly “network based location” you have to agree that Google collects your location data “to improve its services”:
So “location history” and ”location report” are meaningless in this context. As it was adressed: Whilest theoretically you should be able to delete all collected data associated with your account, first Google claims here it's “anonymous” (so not connected to your account) – and second, noone except Google can tell if “deleting” really means the data was “destroyed”. Different reports rather suggest otherwise, as e.g. a German tech magazin found out: They decided to delete all backups from Google servers. Weeks later, they wanted to test a new device, and configured it with the very same account. Miraculously, that device immediately connected to their WiFi. How can that be, with the backups deleted? (Find the full article here, in German).
Will you miss anything when switching to GPS only?
Depends on your “usage profile”. When no satellites are in sight, you would be unable to find your location. That could be e.g. with heavy clouds, or inside buildings – where otherwise a “coarse location” based on WiFi APs and mobile cell IDs would be possible. But there are ways to get that functionality back without Google: The OpenCellID project is aimed to provide the data, and there are several connected apps – including alternative open source network location providers. But to use those, your device must be rooted, as these services must be installed as system apps. You can find out more in my Android without Google series.
Should you use alternative maps?
That's pretty much opinion-based. I do so, but not solely for the reasons you mentioned. There are alternatives (part 2 of the just mentioned series names some). Furthermore, let me quote Dan here:
The answer to "Is X secure?" is always "secure against what?" What are you worried about? It's secure in the sense that it won't let people run malware on your phone, but that's probably not what you're thinking of.
IMHO you cannot avoid completely that some services collect data on you. You can minimize it, you can make it harder for them – and you definitely shouldn't offer your data on a silver tablet. It's always balancing privacy and convenience. So how much privacy you can gain, depends on how much convenience you're willing to give up.
Not only if you're absolutely paranoid (correct: today that's called “informed”), a custom-rom can offer you advantages. As those usually ship without all the pre-installed bloat, your battery will last much longer (on my “Android device without Google”, 3 times longer: 4,5 days, and still 20% left). In this context, I have to mention Replicant – which is aimed at absolute open-source. Unfortunately, only few devices are supported.