Word metric on a finitely generated subgroup versus the word metric of its finitely generated parent
Yes you have $d_1(x,y) \geq \ d_2(x,y)$ in the above set up but in general there is not a ton to say. As mentioned in the comments this is this idea of distortion which measures certain aspects of how the subgroup $H$ fits inside the group $G$. If you would like you can look at this blog post which discusses some of the ideas and defines this distortion functions, which intuitively compares the intrinsic geometry of the subgroup(its own word metric) and how that subgroup fits inside the full group.
A simple example comes from a Baumslag-Solitar group $G=BS(1,2)= \langle a,t \mid tat^{-1} = a^2 \rangle$(discussed in the above blog post, which you should look at -- it has pictures). Consider $H=\langle a \rangle$ in $G$. Well
$$t^nat^{-n}=t^{n-1} t a t^{-1}t^{-n+1}=t^{n-1} a^2 t^{-n+1}= (t^{n-1} a t^{-n+1}) (t^{n-1}a t^{-n+1})= \dots =a^{2^n} $$
which gives that $d_1(1,a^{2^n})=2^n \geq 2n+1 \geq d_2(1,a^{2^n})$ which is a pretty big difference in the geometry.
Now sometimes you can say more, although normally "coarse-ify" things up to some sort of equivalence so that the choice of generating sets does not change the answer. For example in hyperbolic groups and CAT(0) groups it is known that abelian subgroups are undistorted/quasi-isometrically embedded.
The word metric generalises to something called a "length function". I'll define this properly below, but really a length function $L$ is a function $G\rightarrow \mathbb{N}$ satisfying the obvious kind of think, like $L(g)=L(g^{-1})$.
A summary of what follows is:
The induced word metric on a subgroup $H$ of a finitely presented group $G$ can be, basically, any computable length function.
The motivation here is a question of Gromov, which asks how bad can the distortion of the infinite cyclic group be when embedded in a group? What follows says that "if you can find a length function then there exists a group with the corresponding distortion".
Definition. A length function on a group $H$ is a map $L: H\rightarrow\mathbb{N}$ such that
- $L(h)=0$ if and only if $h=1$;
- $L(h)=L(h^{-1})$ for all $h\in H$;
- $L(h_1h_2)\leq L(h_1)+L(h_2)$ for all $h_1, h_2\in H$;
- there exists some $\lambda>0$ such that for all $r\in\mathbb{N}$, the set $\{h\in H\mid L(h)\leq r\}$ contains at most $\lambda^r$ elements.
For example, the word metric defines a length function on any finitely generated group. Moreover, if $H$ is a subgroup of a finitely generated group $G$ then the word metric of $G$ restricted to $H$ is a length function on $H$
Definition. Let $H$ be a finitely generated group. Two length functions $L_1$ and $L_2$ on $H$ are strongly equivalent if there exists some $C>0$ such that for all $h\in H$, $$\frac1CL_1(h)\leq L_2(h)\leq CL_1(h).$$ "Strongly equivalent" is really the same as "equivalent", as changes in generators will alter the length function, but only up to strong equivalence.
We then have the following theorem, which is Theorem 2 of Ol'shanskii, On the distortion of subgroups of finitely presented groups, Matematicheskii Sbornik, 188 (1997) pp. 51-98.
Theorem. Let $H$ be a finitely generated group and $L:H\rightarrow\mathbb{N}$ a computable length function on $H$. Then $H$ embeds in a finitely presented group $G$ such that $L$ is strongly equivalent to the word metric of $G$ restricted to $H$.
To repeat what I said above, this theorem roughly says that the induced word metric on a subgroup $H$ of a finitely presented group $G$ can be, basically, any "nice-enough" length function.
Of course, "nice-enough" is not necessarily that nice. For example, the Ackermann function is recursive*, and hence computable. There are free groups which embed into "nice" groups with Ackermannian distortion! See the paper W. Dison, T. Riley, Hydra groups, Comm. Math. Helv., 88 (2013) pp. 507–540 (arxiv).
*but not primitive recursive, which is why it was introduced.