One of the simplest and most intuitive books covering cointegration is Applied Econometric Time Series by Enders.
It would cover both Engle-Granger and Johansen (although not in so much detail).
Another tried and true way of learning it is to go to the Eviews Help Manual. It has grown over the years and now is over 1000 pages. I used it when it was maybe 300 and learned alll the basics of pretty much all the econometrics I use today in that manual. It has a very recipe-book style, of course using eviews. Examples will involve the author testing for unit root in individual series, combining to test for cointegration and testing the numbers of cointegratin vectors. I'm not 100% sure they cover Engle-Granger (or instead just go for the easier to use one-step Johansen method) but it is a great book nonetheless.
Engle Granger boils down to:
- Test each series to check I(1)--brownian motion, or I(0)- white
noise using Augmented Dickey Fuller (ADF) test. Must be I(1).
Do regression. Check residuals, run ADF test. They must be I(0) -
stationary/white noise....i.e., they gotta cross zero lots of times.
If you pass both, you have cointegration.
To fit the Error correction framework, you take you cointegrated residuals $\epsilon(t)$ and lag them and do a regression, i.e.,
$$\Delta x(t) = a \cdot \epsilon(t-1) + b \cdot \Delta x(t-1) + c \cdot \Delta y(t-1) + .... + \text{resid}$$
and do the same for the other series.
Since ADF has arbitrary lags, sometimes people automate this using a AIC or BIC in steps 1 and 2.