By a result of B.H. Neumann and H. Neunmann (Neumann, B. H.; Neumann, Hanna Embedding theorems for groups. J. London Math. Soc. 34 1959 465--479.), every countable solvable group of class $c$ embeds into a 2-generated solvable group of class $c+2$. For locally finite groups one can use the following construction. Consider the group $S_\infty$ of finitary permutations of $\mathbb N$ (all permutations with finite support). It is generated by transpositions $(1,2), (2,3),...,(n,n+1),...$. The shift $n\mapsto n+1$ induces an injective endomorphism of $S_\infty$ into itself. Consider the (ascending) HNN extension of $S_\infty$ corresponding to this endomorphism. The resulting group is elementary amenable, 2-generated, and contains all locally finite countable groups as subgroups. I do not know of any results about embeddings of countable non-elementary amenable groups into finitely generated ones.
By a result of B.H. Neumann and H. Neunmann (Neumann, B. H.; Neumann, Hanna Embedding theorems for groups. J. London Math. Soc. 34 1959 465--479.), every countable solvable group of class $c$ embeds into a 2-generated solvable group of class $c+2$. For locally finite groups one can use the following construction. Consider the group $S_\infty$ of finitary permutations of $\mathbb N$ (all permutations with finite support). It is generated by transpositions $(1,2), (2,3),...,(n,n+1),...$. The shift $n\mapsto n+1$ induces an injective endomorphism of $S_\infty$ into itself. Consider the (ascending) HNN extension of $S_\infty$ corresponding to this endomorphism. The resulting group is elementary amenable, 2-generated, and contains all locally finite countable groups as subgroups. I do not know of any results about embeddings of countable non-elementary amenable groups into finitely generated ones.