Just on top of my head, I can think of the following themes, all having in common that they can be approached with a minimal knowledge and that there exists a continuous path from them to current research (those with a star could be linked to real analysis):

Quadratic reciprocity*, higher reciprocity laws (starting with the biquadratic character of 2), sample cases of Fermat's last theorem, primality testing*, the last entry of Gauss diary, counting solutions of polynomial equations modulo various primes*, factorization of Fermat and Mersenne numbers, representing integers by quadratic forms, representing integers by sum of squares*, classifying integral quadratic forms, cyclotomy...

Generally speaking, I think perusing the table of content of Gauss' Disquisitiones is a very good source of inspiration for projects like that.