Adjusting for confounding in linear regression model I am wondering how would the slope and intercept change after adjusting for a confounder factor. After adjustment, would the slope be lower, or higher, and the value for the intercept?
Is there any other important numerical change when adjusting for confounding? (in standard errors, etc.)
Thank you!

The "omitted variable bias formula" as stated in Stock and Watson, Introduction to Econometrics (also available at https://www.econometrics-with-r.org/6-1-omitted-variable-bias.html) says that the slope coefficient of a regression of $y$ on $X$ only converges to the true value $\beta_1$ plus a term associated with the correlation of what is not included in the simple regression and hence left in the error term, $\rho_{Xu}$,
$$
\hat\beta_1 \rightarrow_{p} \beta_1 + \rho_{Xu} \frac{\sigma_u}{\sigma_X} 
$$
As @whuber said, depending on your causal question and what is left in the error term and how it is associated with the variable of interest, both directions are possible.
Here are two answers discussing a specific example:
Is it true that an estimator will always asymptotically be consistent if it is biased in finite samples?
Observational vs quasi-experimental design?
The standard errors will be affected, too. For one, all versions of these (homoskedastic ones, heteroskedasticity-robust ones etc.) make use of residuals, i.e. differences between the dependent variable and fitted values. Now, if you compute fitted values from a different regression, the residuals and hence standard errors will be different, too.
In more detail, the proofs of consistency of the standard errors require that residuals are computed from a consistent estimator of $\beta$, so that omitted variable bias will also lead to inconsistent standard errors.
