As an Apostate
Get buy-in from the party, and hide your powers. Surviving as an apostate mage means making sure that the people who care that you're a mage don't know, and the people who know that you're a mage don't care. If the rest of the party is aligned the the Chantry, do not play an apostate mage, full stop. If the rest of your party is independent of the Chantry (gray wardens, mercenaries, soldiers), then they might be willing to put up with an apostate if you keep them alive.
If you have a party that is willing to work with you, the next step is to have a cover story for what you do for them that isn't magic. When the Templars come knocking, you're an herbalist, a tracker, a dealmaker, logistics officer, or anything else that makes sense and isn't a mage.
As a Dalish Elf
While technically apostates (and often blood mages), Dalish Keepers can afford to be a bit more open about their existence. As with other apostates, you might have a hard time with a party that is aligned with the Chantry, but the complexities of politics mean that they could find an excuse for working with you if they need to (say, because they're in over their heads and need a healer). You won't have to hide your power from them, but you might have to keep them at arms length.
As a Gray Warden
As Isaac answered, Grey Wardens have political immunity to all sorts of crimes, since they're basically dead men walking anyway. If you can find a Grey Warden willing to conscript you (or play a Grey Warden from the get-go, if your DM allows), the Chantry won't be able to touch you.
As a Tevinter
The Tevinter Imperium has politics much more friendly to mages. If you play a Tevinter dignitary (or a vacationing noble brat). Just as a Dalish Elf, you'll be able to fall back on Diplomatic protection if anyone goes after you just for being a mage, but be careful that you don't push this too far, as it really depends on your GM how far this diplomatic protection could actually go.
As a Circle Mage
Your GM's claim that a first level mage would not be let out of the Tower does not ring true to me. As long as you've passed your Harrowing (which I assume comes before first level, though your GM may rule differently), you are a full fledged mage, and could be sent out by the Circle to assist on whatever mission of import the party is doing. You wouldn't have freedom to do whatever you like, and the Circle would keep a phylactery so they can track you, but you'd still be able to play the game.
The above represents my interpretation of the setting, and the options that I would present a mage player, were I GMing. All of these require the GM's cooperation, and you should make sure that the GM and the rest of the party are on the same page before you go forward with any of these concepts.
If you're playing in a region (or interpretation of the setting) where Dalish are treated as enemies, or Tevinter is not respected, those options would be closed to you. Likewise, if someone in your party wants to play a Templar, you probably cannot be an Apostate.
Lastly, if another party-member decides to instigate something: don't lay down and take it. Isolate them (the character, that is) and stand up for yourself. Make it clear that there are two sides, and give the rest of the party a reason to take yours. (Or, of course, if intra-party conflict is not the norm at your table, talk to the player OOC)