After the erection of the Mishkan, the Luchos were kept in the Aron therein. But there is an issue of timing. Moshe received the Luchos shortly after leaving Egypt, but the Mishkan was only erected a year later (e.g., Taanis 29a). (We received the second set of luchos on Yom Kippur, the 10th of Tishre. Tradition has it that the Mishkan was finished on the 25th of Kislev.) Where were the Luchos kept for that first year?
Rashi on Devarim 10:1 explains that when Moshe was commanded to hew the second luchot, he was also commanded to build a wooden ark (not the Aron that Betzalel built). This ark was used until Betzalel built the Aron (he was only told about building it 40 days later, and then it took some time, because he build the Mishkan first). Afterwards, when the Jews went to war, the ark that Moshe built was taken with them.
While everyone agrees that Moshe's ark was used unitil Betzalel's Aron was built, there is a disagreement about whether Moshe's ark was still used after Betzalel's Aron was built. You can see a discussion about this here. Some of the opinions include:
The Rashi on Devarim 10:1, mentioned above.
the Ramban 10:5, who disagrees with Rashi, and says Moshe's ark was only used until Betzalel's was built.
Tosafot (Eruvin 63B), who is of the opinion that there were two Arks, but says that when King Shlomo built the Beit Hamikdash both Luchot were placed in one Ark (according to the opinion that the broken Luchot were also in the Ark).