I am going to assume the monoids have units, otherwise slight modifications are needed.
$Mon$ and $AbMon$ have all small limits and all small colimits, $F:Mon\to Cat$ and $F:AbMon\to Cat$ preserve all limits that exist in $Mon$ but preserve only a tiny handful of colimits (even hardly preserve binary coproducts). Many of the properties of $Mon$ and $AbMon$ can be deduced from general properties of monadic categories as each of these categories is monadic over $Set$.
Constructions of limits in each of these categories is pretty much straightforward: Arbitrary set-indexed products are constructed 'as in $Set$' and equalizers are given by a suitable subobject, as usual.
Colimits are a different story. Clearly binary coproducts can't be expected to be preserved since the coproduct in $Mon$ (or $AbMon$) of any two objects will be a monoid again while the coproduct of two categories is almost never a monoid but rather a category with at least two objects.
Coproducts in $AbMon$ are related to coproducts in $Ab$ (the category of abelian groups) while coproducts in $Mon$ are related to coproducts in $Grp$ (the category of groups). The latter construction is not so trivial and tends to manufacture huge coproducts from small ones (in group theory the coproduct of two groups is often called the free product or the amalgamated product).
In general the inclusion $AbMon\to Mon$ preserves limits but not colimits (or even binary products). For instance, if $M$ is the trivial monoid with one element then $M \coprod M$ in $AbMon$ is a monoid with four objects, $m_1, m_2, e, m$ where $m_1,m_2$ are the copies of the object in $M$, $e$ is a newly conjured identity element, and $m$ is the freely conjured product $m_1m_2$. In contract, the coproduct $M\coprod M$ in $Mon$ is countably infinite.