A smooth del Pezzo surface $X_d$ of degree $d\geq3$ has very ample anticanonical divisor $-K_X$. It is also the blowup $X\to \mathbb P^2$ in $k=9-d$ points $P_1,\ldots,P_k\in\mathbb{P}^2_{x,y,z}$ in general position, so we have that $-K_X=\pi^*\mathcal{O}_{\mathbb P^2}(3) - E_1 - \cdots - E_k$. Therefore you can write down the anticanonical embedding of $X$ by writing down a basis for the linear system of cubic polynomials in $x,y,z$ that vanish at $P_1,\ldots,P_k$. For example if $d=6$ then, up to a linear automorphism of $\mathbb P^2$, we can assume that the three points we blow up are $(1:0:0)$, $(0:1:0)$ and $(0:0:1)$ and the seven-dimensional linear system of cubic polynomials vanishing at these points has a basis given by $xyz,x^2y,xy^2,y^2z,yz^2,xz^2,x^2z$, which are the coordinates $b,a_1,\ldots,a_6$ in Will Sawin's comment above. (This also follows from the description of $X_6$ as a toric variety.)

For each $d$ this will give you an embedding $X_d\hookrightarrow \mathbb{P}^d$ of codimension $d-2$ where the $X_d$ is cut out by quadratic polynomials. (For $X_9=\mathbb P^2$ it gives the third Veronese embedding.) However there are many simpler embeddings that one can work with if you want to work with explicit equations. For example $X_6$ can also be described as a smooth hypersurface of bidegree $(1,1,1)$ in $\mathbb P^1\times \mathbb P^1\times \mathbb P^1$.