Not sure if this counts for the OP's purpose, but I started with Hungerford's Abstract Algebra: An Introduction. The first 3 chapters go like so:

1. Arithmetic in $\mathbb Z$ revisited
2. Congruence in $\mathbb Z$ and modular arithmetic (i.e., $\mathbb Z_n$)
3. Rings

Etc.

Not sure if this counts for the OP's purpose, but I started with Hungerford's Abstract Algebra: An Introduction. The first 3 chapters go like this:

1. Arithmetic in $\mathbb Z$ revisited
2. Congruence in $\mathbb Z$ and modular arithmetic (i.e., $\mathbb Z_n$)
3. Rings

Etc. The motivating thesis is explicitly laid out in the Preface and a Thematic Table of Contents -- that it's better to go from well-known concepts to more abstract in fairly small gradual steps (i.e., the reverse of OP's experience with Artin).

Ironically, I now look at a book that starts with groups enviously for its brevity and elegance (rings as a combination of two groups plus distribution), but it seems pretty easy to believe that Hungerford's approach is less of a shock-treatment to students.