Are there going to be "swamp druids" or "desert druids" in your group too? I'm pretty sure, unless the setting is specifically a desert setting or the DM tells you you're starting in a town in the mountains (in which case your bigger concern is how to you explain your being there?) or whatever...but my point is (sorry for the digression) that I would think it will be assumed you are a "forest druid" without you saying anything.
The suggestions above are all good ones (I especially like the phrases/speech suggestion). I'd only add, maybe, that your description of your appearance should tell them everything they need to know. Are there feathers of hawks or owls or colorful songbirds tied to your staff/spear/cudgel? Or on a headband or just tied to your hair...Are there oak leaves and sprigs of holly or mistletoe jutting out of pouches, belt, or wrapped up in your hair (which I think has been done to DEATH, but it beats the twigs and antlers everyone seems to have these days). Grey-green or green-brown leathers/outfits/cloaks say "woodlands" pretty well.
How are you going to describe how your character actually looks? That should do most of the work.
If someone asks you where you're from, you can say "The southern woodlands or such-n-such forest" or whatever. Other than that...there should be no reason for you to ever bring up/say outright, "I'm a forest druid."