Would both VPNs and the TOR network put a stop to this surveillance?
Which is the better option for every day use (least speed reduction)?
If VPN is the best option what are favourites to use (again, with focus on privacy and speed)?
Recommendations for a specific product are considered off-topic here, and the trouble with VPNs anyways is that we can't verify that any are good - we can only tell you which ones are known to be bad.
At a high level, VPNs and tor are very similar: rather than your internet traffic leaving your house with a destination address of the server you want to contact, you send it instead to another server, which proxies the request to the real destination. In both cases the connection between you and the proxy server is encrypted, which means the most likely place the government would be able to observe your traffic is now cut off for them. Do note, though, that they will be able to know that you're using a method to avoid them. (That's one of the arguments for pushing encrypted traffic by default and getting as many people as possible using tor, so that this no longer becomes an interesting signal to eavesdroppers).
However, there is a really important difference between VPNs and tor, and that has to do with trust. The VPN knows everything that you do, and you have to trust them to keep that a secret. Cheap VPNs are known to sell browsing history for advertising purposes. If the VPN provider is located in the UK, they almost certainly will provide customer logs on government request; even if they aren't, they might, as long as they don't think it will get out that they did so (since that would be bad for business). And even a well-intentioned VPN is a great target for an intelligence agency to hack into.
The Tor protocol, on the other hand, is specifically designed to retain privacy when used with untrustworthy nodes.* The exit node knows the same information as the VPN provider about the request (who it's going to, what its contents are, what the response is), but has no idea who is making that request. The entry node knows that you're making a request, but has no idea what it is. And the middle node knows essentially nothing.
Although this isn't directly related to your stated privacy goals, tor also rotates through exit nodes every five minutes, which helps prevent a website operator (or advertising network) from building a profile on you by IP.
You should always use https connections with tor when possible. There are privacy reasons to do so that are best explained with this diagram from the EFF (this applies to VPNs, too), and also security reasons, given that you're essentially introducing a man-in-the-middle into your requests.
In short, tor provides distinctly better privacy than VPNs, both in general and for the specific things you're worried about.
* There are some correlation attacks when an attacker controls both the entry and exit nodes in a circuit, so go run your own node to add diversity!