A perspective convert is not obligated to keep any extra mitzvos for it. Today it is common for a beis din to independently decide as a matter of policy to require the show of commitment of a candidate practicing while readying him/herself for the actual conversion. After all, the person could try the lifestyle and find it leaves them cold, avoiding a possible mistake. But at least the person will be in practice, and less likely to forget to wash hands before eating or make a berakhah, or the like. But it's not a halachic duty on the person, as there is no special halachic category of being "pre-conversion". Nor is such observance a mandatory part of the conversion process.
In fact, a problem for prospective converts are the one or two mitzvos that non-Jew ought not keep.
For example, Shabbos. Hashem describes Shabbos as "an eternal sign between Me and the Jewish People." A non-Jew observing Shabbos is somewhere between theft and pseudo-adultery. For men, one clever solution is to wear tzitzis. For a Jew, who is obligated in wearing tzitzis, the tassles are considered subsidiary to the garment, which is worn on Shabbos. For a non-Jew, the tassles are unnecessary in terms of attire, so they are being carried no less than if they were in the person's pocket. There is also a question whether an eiruv works for a non-Jew, so carrying in an eiruv would be something the prospective convert can do without standing out amongst his shul mates.
The limits of Torah study for a non-Jew is more complex, with many extant rulings. Obviously they're allowed to learn the things they need to know to be good Jews day one -- applicable laws, the fundamental beliefs (which the Chovos haLvavos would also call "applicable laws", the title of the book is "Duties of the Heart"), and likely Tanakh. Paritcularly someone living in a Christian culture, where Tanakh is respected, if under a different name and in a translation that has a number of alterations. But because there is a variety of rulings, the applied rules would depend on the beis din.