Your first sum can be expressed as a $q$-digamma function (the Lambert series being a special case of this); more precisely, we have (using sweetjazz's form):
$\sum_{\ell=0}^\infty \frac{xy^\ell}{1-xy^\ell}=\frac{\psi _y^{(0)}\left(\frac{\log\,x}{\log\,y}\right)+\log(1-y)}{\log\,y}$
The second series requires a bit more work; first we note that
$\sum_{k=1}^\infty{\frac{x^k}{1+y^k}}=\sum_{\ell=0}^\infty (-1)^\ell \frac{xy^\ell}{1-xy^\ell}=\sum_{\ell=0}^\infty \frac{xy^{2\ell}}{1-xy^{2\ell}}-\sum_{\ell=0}^\infty \frac{xy^{2\ell+1}}{1-xy^{2\ell+1}}$
The two series in the last expression now bear some resemblance to the series for the $q$-digamma function. In fact, we have the following:
$\sum_{k=1}^\infty{\frac{x^k}{1+y^k}}=\frac{\psi_{y^2}^{(0)}\left(\frac{\log\,x}{2\log\,y}\right)-\psi_{y^2}^{(0)}\left(\frac12\left(1+\frac{\log\,x}{\log\,y}\right)\right)}{2\log\,y}$
It looks to me that both series can alternatively be expressed as (products/ratios of) Jacobi theta functions (and their derivatives), but I haven't tried that route and my knowledge of modular identities is not up to snuff...