How do I tell if I'm playing on a Valve server?

You can tell that you're playing on a Valve server in two ways.

One way is to look at the server name and see if the name of the server is displayed on the leaderboard (default key Tab) as being a Valve server. Usually the name format of this will be Valve [gamemode/event] Server ([server location] srcdsXXX #XX.

Another way, if there is any sort of event going on that includes contracts, is to see if a contract is not 'inactive' on the server. If a server you're on is a Valve server, the contract progress bar will show up on the right side of your screen.

Otherwise, the contract will still be shown as inactive regardless of whether or not you're on the right map with enough players.

As for a way to connect to them more often, you can use the Play Multiplayer menu to find them by setting the search settings to look for Official servers.

You can also use the Show Servers button to see a list of available servers meeting any criteria you set this way.

EDITED

As of 07/07/2016 (Meet Your Match Update), Quickplay (the standard method outlined above) has been replaced with Casual Mode matchmaking, effectively meaning that all servers joined through matchmaking will be Valve Servers. Community servers can instead be accessed through a separate option using the legacy server browser.