Are there any simple and yet secure encryption algorithms?
A simple block cipher would be Threefish (p. 11-13). It's a bit more complicated than RC4 or RC5 yet doesn't drive you insane with seemingly random design choices. It is presumably secure and was designed by experts but has not yet been reviewed extensively, so it shouldn't be used in sensitive applications yet (consider it an exercise).

You'll be able to implement the key schedule and the encryption/decryption functions, and if you use the PDF I linked above, it will be an excellent exercise in following a specification.

If anything, it's the technical details surrounding the implementations that can take you forever to fix, such as endianness concerns in particular (this isn't too much of a hassle for block ciphers but it did get in my way a few times when implementing hash functions).

PS: same name lol