What can we say of a group all of whose proper subgroups are abelian?
The finite case was settled by Miller–Moreno (1903) and described again in Redei (1950). The infinite case is substantially different, due to the existence of Tarski monsters. However, the case of non-perfect groups is reasonably similar to finite groups and is handled in Beljaev–Sesekin (1975), along with some more general conditions. Generally speaking, infinite groups are not very similar to finite groups in these sorts of questions, but if you restrict to (nearly) solvable groups, then things are a bit better. Nearly solvable in this case means "not perfect" and for quotient properties, often means "non-trvial Fitting/Hirsch-Plotkin radical". In other words, for subgroup properties you want an abelian quotient, and for quotient properties you want a (locally nilpotent or) abelian normal subgroup. More current research along the lines I enjoy generalize "abelian" rather than "finite"; a reasonable framework for this is given in Beidleman–Heineken (2009).
Miller, G. A.; Moreno, H. C. Non-abelian groups in which every subgroup is abelian. Trans. Amer. Math. Soc. 4 (1903), no. 4, 398–404. MR1500650 JFM34.0173.01 DOI:10.2307/1986409
Rédei, L. Die Anwendung des schiefen Produktes in der Gruppentheorie. J. Reine Angew. Math. 188, (1950). 201–227. MR48432 DOI:10.1515/crll.1950.188.201
Beljaev, V. V.; Sesekin, N. F. Infinite groups of Miller-Moreno type. Acta Math. Acad. Sci. Hungar. 26 (1975), no. 3-4, 369–376. MR404457 DOI:10.1007/BF01902346
Beidleman, J. C.; Heineken, H. Minimal non-F-groups. Ric. Mat. 58 (2009), no. 1, 33–41. MR2507791 DOI:10.1007/s11587-009-0044-2
If $G$ is a group with all its proper subgroups abelian, then $G$ itself may not be abelian. A perfect counter example is group $D_6$, i.e. $S_3$.
If $G$ is a group with all its subgroups abelian, for sure including itself and trivial one, then yes, $G$ itself is also abelian. You can easily check commutative of arbitrary 2 elements by generating them into a subgroup of $G$.
Hope that helps.