This is probably too late for you now, but I'll provide some info in case anyone else needs it.
I define a (real space) lattice using
$$\mathbf R_{[m_1,m_2]}=m_1\mathbf a_1+m_2\mathbf a_2.$$
Here $\mathbf a_i$ are the lattice basis vectors. I used two dimensions but this is easily generalized. I can now create lattice of delta functions using
$$\rho(\mathbf x)=\sum_\mathbf {R}\delta(\mathbf x-\mathbf R),$$
where $\mathbf R$ is in the lattice. I can easily create more intricate patterns by convolving this lattice with some other compact function: $\rho'=f*\rho$. Here $f(x)$ is a function that fits within one unit cell and is zero everywhere else. The result is $f$ repeated at every unit cell.
Let's now look at the Fourier transform of this lattice:
\begin{align}\mathcal F_{\mathbf k}\left[\rho(\mathbf x)\right]&=\int\mathrm d \mathbf x\, e^{i\mathbf k\cdot\mathbf x}\rho(\mathbf x)\\
&=\sum_{\mathbf R}e^{i\mathbf k\cdot\mathbf R}
\end{align}
Without a proof, we can see that this sum goes to infinity whenever $\mathbf k\cdot \mathbf R=2\pi$. It will go to zero everywhere else. So, relying on intuition, the result is again a lattice of delta functions, with a delta function or every k-vector where $\mathbf k\cdot \mathbf R=2\pi$. This happens for all points of the form
$$G_{(n_1,n_2)}=n_1\mathbf b_1+n_2\mathbf b_2,$$
where $\mathbf b_i$ are the reciprocal basis vectors, defined using
$$\mathbf a_i\cdot \mathbf b_j=2\pi\delta_{ij}$$
To calculate these vectors explicitly, it is insightful to collect these basis vectors into a matrix:
\begin{align}
&A_{ij}=(\mathbf a_i)_j\\
&B_{ij}=(\mathbf b_i)_j,
\end{align}
which you should read as the $j$-th component of the $i$-th basis vector. Our definition of the basis vectors can now be written as
$$AB^T=2\pi 1\!\!1$$
such that $B$ is given by
$$B=2\pi (A^{-1})^T.$$
Once you have the reciprocal basis vectors, drawing the reciprocal lattice should be straightforward. Is your real lattice 3D? Then the reciprocal lattice is as well. Drawing in 3D by hand might be hard, in that case you could draw them using any software that can draw 3D points: Python, geogebra, Mathematica, Matlab, Desmos. Make an array of points with $n_1, n_2, n_3$ varying over a range of numbers and then plot those. Try to keep the numbers small, because the number of points quickly blows up.
