Let me also say that YES, I know Google makes an extra point to make every single service mobile compatible on the web. But I'm talking about outside of Safari.
Also, I have an iPod Touch 3G which I don't think can be jailbroken, so all the apps I talk about are from the App store.
(Sidenote, I am cheap, so I probably won't end out talking about any paid apps...)
I'm not really a fan of Apple software (including Apps), but I've gotta say that I'm ok with the Mail app. It's really all that I wanted in an app....it syncs both ways, it has push notifications, it has Draft and Sent Mail folders, it syncs all your tags, and (best of all), it actually downloads the mail. That's great because I have an iTouch and not an iPhone, so I can check my e-mail, go to a place that doesn't have wi-fi, then pull up my Mail and look at the e-mail again.
HOWTO: Fortunately, Apple's default Mail app already has extreme support for syncing with Gmail. All you have to do is go under "Settings", go to "Mail, Contacts, Calendars", click "Add Account", and a big giant "Gmail" option greets you. From there, it's self explanatory.
Apple's Calendar app does not impress me as much as its Mail app. It can sync both ways with a Google Calendar, but it lacks features that Google has, in my opinion. Every view (List, Day, and Month) seems crowded, repeating events don't have options like "Every Tuesday and Thursday", everything is incredibly Grey, and (what I wish for the most) there's no "Week" view. I really wouldn't mind it much if it had this feature that Google has, but it doesn't. All I'm asking is to change from "Day" to "Week" when you flip the iPod sideways, but alas. That feature really helps me see my week at a glance in Google.
HOWTO: To add a Calendar simply, go to "Settings", go to "Mail, Contacts, Calendars", click "Add Account", "Other", and finally "Add CalDAV Account". Set the "Server" to "www.google.com", the "Username" to your full e-mail address (ie. "firstname.lastname@example.org"), "Password" as your password, whatever description you want to be shown in the Calendar App, and you should be good.
This is really nice because it's accessible through other apps. It's the address book Mail uses to lookup contacts, you can lookup contact's addresses in the Maps app, and you can even lookup people from it in 3rd party apps like adding professors in iStudiez.
All in all, the app is very well constructed, although very very simple. The only thing that really sucks is the lack of groups.
HOWTO: When you connect your iPod/Phone(/Pad?) to iTunes, click on its name in the sidebar, then click the "Info" tab (at the end of "Summary", "Applications", "Music", etc). You should see a section called "Contacts". Check the box for "Sync Contacts" and select "Google Contacts"Enter in your username and password, and you should be good.NOTE: Check your iPod and see if it worked, it didn't for me. If it does work, you're all set! If it doesn't, go to "C:\Program Files\Common Files\Apple\Mobile Device Support\bin" and find the file named "gconsync.exe". If you're on Windows XP, try set the file to run in compatibility mode for Windows 98. If you're on Windows Vista, set it to run this file as administrator. Sync your iPod/Phone(/Pad?) again, and this time it should work! It did for me, anyway. (I'm on XP, btw).
Gmail, Calendar & Contacts.
Another way to sync all three of the ones I talked about above at once is to use Google Sync. Google Sync uses Microsoft Exchange to sync multiple accounts: mail, calendar, and contacts. This one requires at least OS 3.0, and it requires Microsoft Exchange Activesync to be enabled. (Whatever the heck that means.) Anyway, Google endorses it, but I personally decided to do the three separate ways I listed above.
HOWTO: Google has an official guide, so check there if you really want to do it, but here's the basic rundown: Select Add a Mail/Calendar/Contact account; choose Microsoft Exchange; fill in your e-mail, leave the domain blank, use your full e-mail address as the username, fill in the password; hit "Next"; enter "m.google.com" as the Server; hit "Next"; choose which of the three you want to sync.
NOTE: I tried doing this for just Contacts and it only synced one way, ie: it downloaded my Google Contacts to my iPod, but didn't upload my iPod contacts to Google. So that's why I went with doing each of them separately.
As far as I know, Apple doesn't have an RSS app, so this one requires a third party one. It's really all about preferences, but I tried to find one that is most like Google Reader itself. I've gotta say that was alot more challenging than I thought.
My old favorite was called NetNewsWire (free). It's not bad, it has everything well divided into collapsible sections and was very straightforward. The only things I really didn't kinda like about it is that it only synced starred or unread items, and you couldn't mark an item as unread.
A new one I just recently found is called MobileRSS (free). The main menu looks cluttered at first, but it's totally customizable. It has everything divided up into separate pages for each folder, which makes it seem a bit "larger", if you will. It has a wonderful black skin that I absolutely love (the blue one is meh...). AND it has the ability to show only updated feeds or ALL. That's one of the things that set it apart from NetNewsWire: you can view all of your feeds, not just the unread items. The way I was using NetNewsWire, I used it to read unread items every now and then, but if I wanted to make a note to read something again later on, I had to star it, and after I read it once, it disappeared. So I still ended out either using GReader in Safari or my computer. MobileRSS is wonderful because it really gives you the full Google experience. You can follow links, e-mail a feed item, add notes, post something to Facebook, Tweet an item, view in fullscreen, view feeds offline, and even unsubscribe to feeds right from the app.
And best of all, it's free. The only "catch"? It's got ads. I use quotes because the ads are totally worth it. This thing has everything I (personally) want in an RSS reader, web or mobile, and all I have to do to use it is see a tiny ad here or there.
But seriously, there are a ton of RSS readers out there in the App store. (Not as many as Twitter apps, but there's a Twitter app for every star in the sky.) So search around if you want. The reason I like MobileRSS is that it's only a Google Reader app, so it's built specifically to work around that, whereas other RSS readers have Google sync as a side (or some don't have two way sync at all), so it's not built for Google.
HOWTO: Pick an app from the App store, install it. Just make sure it says that it supports syncing both ways.
I've searched around for this, but mostly multi-protocol IM clients. I've tried IM+ Lite, Nimbuzz, and Fring, all which I'm not extremely fond of. (Nimbuzz is my favorite of the lot, though.)
I haven't really used this one as much, but the three good ones I can remember are eBuddy, Quicksilver, and maybe Mundi IM. Quicksilver itself does practically everything gTalk can do.
The most disappointing thing about it is that you really can't stay connected unless you are in the app. At least that's what I've found.
HOWTO: Search through the App store.
That's the ones I've found and use. Here's other ones that I haven't been able to work out yet.
I've searched around a little bit and I can't find a good free app for this. There used to be a good one called BlogPress Lite, but it's been removed, and now there's just the paid one. If I blogged more, I'd say the $3 is totally worth it, since I've heard amazing things about BlogPress, but oh well. Maybe I'll convince myself to do it eventually.
The closest thing I can find is called Documents by Savvy Soda. There's a ton of apps that let you view Google Docs, but few that let you edit them. Documents does, but the only catch is that you have to download the document to work on it, then you can upload it to Google when you're done. Not really syncing, which really kinda defeats the purpose.
The thing that sucks about this is that even the Google Docs web app sucks. You can't edit formulas in the spreadsheet, which defeats the entire purpose.
I know it's fairly new, but there are literally under 5 apps I actually see in the app store specifically about it. Only one of them is free, and all it shows is nearby Buzzes, not even who you are following.
[UPDATE 7-6-10] If you are able to/want to jailbreak your iDevice, GV Mobile is available in Cydia.
Really the only thing to sync for this would be bookmarks, but it unfortunately doesn't really look like you can do that yet. It seems odd, since I'm fairly sure the Maps app is by Google, but you can't sync.