I bought the series this weekend and I wanted to make sure that installing the games wouldn't require any other external tweaks/mods to function properly.

The only mod I installed was the Children patch for Fallout 1 since I heard that the kids appeared as ghosts or something, but otherwise, is there anything else I should install that hasn't already been included in the Fallout Series by GOG in 2013?

Thanks for any help!

Awesome! Thanks a bunch!

I disagree, the GOG version includes mostly official patches, which left hundreds of bugs open. These are taken care of in the major fan made patches, which are widely recommended, even by GOG (see the sticky posts in the forum).
GOG doesn't apply them because there are different ones and they're mutually exclusive. Thus providing free choice for the player.

Don't start a game without fan-made patches, because you can't use any vanilla savegames afterwards.
And even then, the game crashes sometimes and tends to corrupt savegame files, so save often and in different slots.

This applies to both, Fallout 1 and 2.

Except that the question was: "does the Original Fallout Series for GOG need any third-party patches to run properly?"
It does not.

The majority of 3rd party fan 'mods' or 'fixes' (not patches) include highly subjective changes or 'enhancements'.

Look, I don't want to argue. This thread is to help Chancholoraq making a decision whether he should install 3rd party patches or not.

Will Fallout 1 or 2 run without 3rd-party patches?
Yes.

Will they run properly?
No. (Please consult the documentation of the major fan made patches for details on fixes.)

Are the big fan made patches more about new content?
Yes and no.
You can usually choose between a version with fixes only (patch) and a version with additional content (mod).

My impression is, that the majority of players are recommending to use at least the fan made patches to get all fixes.
Whether you like the content changes and additions or not, is a matter of taste. Often it's very close to what was originally intended (e.g. like the Children patch). I made good experiences with the mod versions.

Examples of popular big patch/mod packs:
Fallout 1
Patch only: FIXT FIXES ONLY Installation
Patch + mod: FIXT FULL CUSTOM Installation
http://www.nma-fallout.com/forum/viewtopic.php?p=840100

Fallout 2
Patch only: Killap's unofficial Fallout 2 patch
Patch + mod: Killap's Fallout 2 Restoration Project
http://www.nma-fallout.com/forum/viewtopic.php?t=63996

I think this actually depends on your definition of "properly".

If you mean, will they run with a minimal possibility of game-ending bugs, then no they don't need patches.

If you take it to mean "will they run as the designers intended them originally" (but couldn't/didn't due to various factors), then no. This is especially true of Fallout 2 since there is a LOT of content in the game that is normally not accessible.

Although I can't vouch for FIXT (since I did play through F1 without patches - although next time I play it I'll definitely add it, as the one I got from GOG was missing children for some reason), I am likely one of many that would recommend at the very least the fan-made patches to Fallout 2, but ideally the Restoration Project for Fallout 2 since the things it adds are things that were "mostly" in line with what the developers were thinking anyway.

Plus F2RP fixes some combat bugs that remained.

At the end of the day, it's up to you how much you do or don't mod - this is just my 2 cents worth. Links to what I would install are in the post above.

Yet you continue to. Absolutely incorrect. I run Fallout and Fallout 2 without any 3rd party fan mods, and they both run flawlessly.

[... snip]

Hickory is right, they'll run just fine without fan-made patches or mods.

However, a lot of the fan-made patches make them better. You've already found the children patch for Fallout 1, which is merely an anti-censorship addition, and I particularly recommend killap's Restoration Project 2.2 for Fallout 2 (even for a first time player). I would not say it includes lots of changes that are "highly subjective." The author has gone to great lengths to limit the scope of the RP to fixing bugs, typos, and functions that weren't working correctly, and reintroducing content that was planned by the original creators of the game but cut due to time constraints. Interplay even includes a version of it on the DVD trilogy versions they sell these days.

But do you have to use any fan-made patches? No.

Children patch was already mentioned and is the most obvious example.

In Fallout 1 there's a quest to find a missing child.
In unpatched GOG version there are no children, so you can't find the missing one, so you can't finish the quest.

There are other examples of a quests that have a bug in them, so you can't finish them if some condition applies.
Such things are fixed in the patches, more critical ones in FO2.

The games are, however, made so you can fail most quests for whatever reason and still finish them.
The critical plot quests work fine in GOG, so yes, you can play the games and don't need the patches.

But that's semantics, I can't see any reason to not include the fan made bug squatting patches.
The restored content, added content, or balancing patches are another thing entirely.