Can you do this with an iPhone?

I want to have photos and only photos that I take with my iPhone sent to the cloud (when did we start calling the Internet “the cloud?”) so that I can easily access them with my laptop . Can I do that and if so, how?

If you have iTunes installed on your laptop you should be able to access your cloud pictures.

Or do it with Apple’s own Photostream.

Flickr app? G+ app? I think iCloud does it automatically?

iCloud photo stream does exactly what you described.
In osx it’s tied in with iPhoto.
On windows it’s tied to a set of user configurable directories. Just install this:

Works great. I take a snapshot with my iPhone and a few seconds later it’s on my pc in the directory I specified for photo stream. Movies shot don’t sync over iCloud, only photos.

There’s no web access to the iCloud photos, though?

Not for anyone other than you. It’s just server-side storage, not an image-hosting service like Flickr.

I use an app called Camera Sync to upload any new photos/videos to dropbox, which I have set to launch automatically ever night after I’m asleep (via a jailbreak app).

I use the Android and Windows desktop versions, and I’ve never had any issues.

Isn’t there a G+ app for iOS? Seems like that does exactly what you want.

I’d also add that SugarSync by default detects (on the Android app) when you have new mobile photos and asks if you’d like to upload them, but there’s an auto sync setting you can enable also.