Requesting to take photographs

So, I will be doing a study abroad in Nagasaki this upcoming fall semester, and I plan to take lots of pictures to document the experience!

If I were to request to take a photo of something right in front of me, would it be more grammatically correct to say

この、写真をとってもいいですか

Or

これが写真をとってもいいですか

I hope I'm on the right track at least... please let me know if there's a better way to phrase the question.

In that regard, is it considered rude to take pictures of your food before eating? I'll be staying with a host family, and I don't want to offend them or make them uncomfortable with pushy requests...

Thanks!

Indirect object.

これ／それ／あれを写真に撮ってもいいですか。

If the object you want to take a photograph of is obvious to the person you are speaking to, then shortening the sentence makes it sound more natural.

写真に撮ってもいいですか。

No need Japanese in this case.
Just showing your camera/video/cellphone with suggestion by a finger both first
then say "OK?"or "Can I?" enough very simple communication.

If would like to say in Japanese
撮影できますか？ Satuei deki-masuka? /formal
撮影して(も)いいですか？ Satuei shite(mo) iidesuka?
写真(を)撮って(も)いいですか？ Syasin(wo) totte(mo) iidesuka?
http://ejje.weblio.jp/sentence/content/%E6%92%AE%E3%81%A3%E3%81%A6%E3%...
写真撮ってもいい？ Syasin tottemo ii? /frank for your friends and same generation
撮っていい？ Totte ii?/super frank for your friends and same generation

If the answer "Yes!(いいよ!,どうぞ!,構わないよ!,好きに撮って!)" at restaurants,stores or public spaces of house/building
but don't use "flash light" as basic manner.
So ask フラッシュを使って(も)いいですか？/Flash wo tukatte-mo iidesuka? before shooting.
(Cellphone small light mostly OK just be careful direction/angle of light)

撮影 means include movie(動画：dou-ga) recording by video cam,mobile pc web cam and cellphopone.
写真 means photograph.
撮る: to-ru(verb) means take a picture/photograph and recording movie both.
http://ejje.weblio.jp/sentence/content/%E6%92%AE%E3%82%8B
So if would like to record a movie
動画(を/wo or で/de)撮って(も)いい？
this を(*) means "for"
this で means "by" or "as"

Anyway,above Japanese copy to input box then click speaker icon.
http://voicetext.jp/

*)
お/を are same sound shorty "o" just distinguished for writting as particle "を".
https://en.wikipedia.org/wiki/Japanese_particles
So を is no need "w" sound and all お won't be particle.

It can be either a direct object or an indirect object (in English grammar equivalent explanation).

If you say:
写真に撮ってもいいですか。
That is like "May I capture (this) INTO a photograph?," so that would be like an indirect object, and would take a "ni."

...and if you say:
写真を撮ってもいいですか。(possibly pointing at the item you want to photograph)
That is like "May I take a photograph?" so it is like a direct object and would take a "wo."　I feel that this one is more often used.

Considering that it will be at your host family, and that you are probably going to be dining with the family members, I would ask quickly and take a photo quickly, so that everyone can proceed to eat :)

(I have heard that some people in restaurants take inordinate amount of time to take photos of the food, from different angles, with different settings, etc. before they start eating, so that the staff feel the food would get cold or that the customers are not really there to enjoy their food but just to take photos... )

As suggested,
写真を撮ってもいいですか。
is the most common and all-around expression we use in Japan. Just say it as you direct your camera or hand to whatever you want to take your picture of. You can use it for items, people, scenery... anything.

Wow, thanks for all the helpful advice, everyone! Y'all are awesome. After learning all the different variations, I'm even more motivated to keep up with my Japanese language studies now.