Take-out in Forest Hills
Most places seem to be on the south side of the Boulevard...all of these are.
Peking Duck Forest - "American" (not downtown Flushing-type) Chinese restaurant that does take-out and delivery
Latin Cabana Express - Cuban basically take-out, but they may do delivery through some service
The Family Restaurant - restaurant that does take-out - not sure about delivery
A & J Pizza - Pizza place that also serves meals and does take-out and deliveries
There are many others, but I don't do much take-out/order in. If I think of More, I will add...
I second the recommendation of La Cabana Express, definitely a great addition to the neighborhood (in fact, I am eating the leftovers for dinner tonite). The Greek place on Austin ( the one upstairs whose name momentarily escapes me) will deliver also. Dh and I did it once and the food that we got didn't travel so well, but a friend of mine did it once for a lunch (stuck to salads and kebabs and the like) and it was very successful. (I've had lunch there a few times and the food is quite good.) IMHO don't waste your money on Peking Duck Forest. It is way overpriced and the food is nothing special. Welcome!
I grew up in Forest Hills... and now I live in Brooklyn, so I understand your predicament.
Mehak is a great Indo-Pak place on right off Lefferts Boulevard on Austin Street (near the great independent movie theater). They do take-out and, I think, delivery. I hear mixed things about Dani's pizza on Lefferts: never tried it. Some people think it's fabulous, others hate it. These are sort of in Kew Gardens, but it's a nice spot (a little less strip mall than Austin Street in FH).
AJ's on Austin is solid. I'd also suggest Sgt. Garcia's for Mexican (haven't been there in a long time, though, so I can't guarantee it). I think they deliver through some overpriced service, but I'm sure they do takeout.
Habana on 70th Road, I think it is, has nice Cuban food. Not sure if they do take-out/delivery.
Dee's Pizza delivers through a service and they do a great gourmet, thin crust sort of pizza.
Forgive the long litany... When I lived in the area I made it my job to find decent food (cause it's not always easy). One more thing: if you're ever jonesing for a food outing (after the winter), take the Q23 from 71st/Continental down 108th Street to Empanadas del Parque and the Lemon Ice King of Corona. Awesome.
Sgt. Garcia is atrocious. It is, by far, the worst Mexican I've ever had. Why anyone would go there when there's a better, cheaper alternative at 5 Burros, I cannot understand.
Latin Cabana Express, different from Cabana, does delivery for good Latin food.
Thai Pot also delivers.
On several occasions I went to Boston Market and took home 1/2 roast chicken with mashed potatoes and cream spinach. I just like this combination. The Fish Store on Austin has prepared foods which I've taken home that just needs reheating. Once I took home a small container of squid in tomato sauce which I added to thin spaghetti which took 5 minutes to cook. They also have salads (tuna, crab, shrimp) that are nice summer meals. I've taken home grilled chicken or steak with rice for $4 at the Asian owned Tex/Mex place on Austin St.
I lived in Forest Hills for almost 5 years. I still get takeout from on occasion on my way home is East Ocean Palace for Chinese Food (Queens Blvd around 76 Ave). I also liked Sato for Japanese, although that's well down Queens Blvd on 65 Ave or so. My favorite spot in the general neighborhood is Salut on 108 Street near 63 Avenue, which is Uzbeki and has great kebabs, hummus, and bread. Peking Duck is ok. Had a horrible experience with Cabana in terms of service so I refused to ever return.