When I was an undergrad studying abstract algebra, we used the second edition of Artin and covered groups first and then rings. Fields, vector spaces, and algebras came later, I think.
I remember finding group theory difficult because of the number of concepts (cosets, group actions, &c) and the large amounts of only-slightly-different notations for things like orbits and stabilizers. Another thing I remember pretty clearly is being confused by the different meanings of juxtaposition in different contexts (e.g. $gHg^{-1}$ vs $ghg^{-1}$ vs $gX$ where $X$ is a set).
I'm wondering whether there are any books or perhaps courses that don't start with either groups or rings and instead pick something more unusual. I'm not a teacher; my interest in this is entirely theoretical.
Based on this question and this question, it seems like teaching groups or rings first is popular, with rings being firmly in second place.
There are other things though that are perhaps reasonable candidates for an introductory algebraic structure.
- Fields -- there are a few ready examples, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, and the finite fields. And there's cool stuff like $\mathbb{H}$ if we're willing to consider skew fields.
- Vector spaces -- row/column vectors with matrices as linear maps and some other examples should be familiar from linear algebra, which is usually taught first.
- Lattices -- lattices are reasonably straightfoward (in my opinion)
And there are some other things that are weird, but also perhaps reasonable choices.
- Monoids or semigroups
- Semirings -- rings without negation.