What resources are available to trace the connections from the Gemara to the Halachic codifications and the reverse? For example, how can someone learning Daf Yomi find the related Halachic rulings in the Shulchan Aruch that were based on that particular page of Talmud, and how would someone studying Mishneh Torah find the source in the Gemara?
The Be'er HaGolah (printed next to the Shulchan Aruch) gives all the sources as listed in Tur/BY. The Biur HaGra gives more thorough sources than the Be'er HaGolah, so you should probably use that one.
There is also, of course, the Ein Mishpat, if you want to go the other way around (Gemara to poskim). It's found at the top of the outer margin of each daf, and it's referenced by k'tav ashurit letters throughout the text of the Gemara.
If you want to learn the codified halachot that are based on the page of the Talmud you're learning, you should definitely check out the Gemara put out by The Halacha Brura and Birur Halacha Institute.
An idea from HaRav Kook, see here:
It was the Rav's desire to publish a new edition of the Talmud, which for the first time, would combine and reunite the study of Halacha and Gemara, which over the years had become two different and independent areas of study...a Talmud which would "return" the basis of Gemara study to the summary of the Halachic conclusions in each and every sugyah, including side discussions which arise in the sugyot.
It is split into two parts, the Halacha Berura and the Biur Halacha. From here:
Halacha Brura: This is printed on the Gemara page and was written by Rav Kook himself. The Rav copied the Rambam's and the Shulchan Aruch's Halachic decisions for each topic discussed on that page of the Gemara. This follows the numbering of the Eyn Mishpat.
Rav Kook also clarifies situations where the Eyn Mishpat is not pointing to the correct place, and sometimes makes notes on what Eyn Mishpat was thinking when pointing to a specific source.
Berur Halacha: Printed in the back of each Gemara, the Berur Halacha summarizes all the different opinions of the Rishonim and Achronim that have impact on the Halachic conclusion of each Sugya, whether they affect the main discussion, or touch only a side point.
The Halacha Berura was done by Rav Kook himself, while the Berur Halacha is done by a committee of Rabbis led by Rav Kook's son (Rav Kook passed away during the editing of the first volume).
I have this Gemara on Berachot and love it. The only possible complaint is that it is not one of the newer prints of the Talmud (only the Rashi is re-typeset). This doesn't bother me for the Gemara itself, but I would prefer the compilation (and retyping) of the Mefarshim found in the back of the Talmud.
b a's answer lists the standard short references from g'mara to halacha and vice versa.
Menachem's answer lists a resource for following a g'mara through to its halachic conclusion.
So I'll do the opposite: mention a resource for following a halacha back to its talmudic source. The best I've found for this is Aruch Hashulchan, who often quotes the relevant g'mara, lists how rishonim understood it l'halacha, often discusses questions on those rishonim, and so on, until the final p'sak.
The new Koren Talmud actually does a very good job at providing not only the Ein Mishpat, but it also gives you the complete Rambam and Shulchan Aruch relevant to particular sugyot.
For the Mishneh Torah, first look at the commentaries printed in the standard editions. However, the work of these commentaries didn't exhaust all the halachos of the Rambam. Some of the gaps have been filled in by R' Chaim Kanievsky's in his Kiryat Melech, which fortunately, is available online.