How can I do a mediation model with one categorical and two continuous variables?

You have to say what variables are where in the model. Is the categorical variable the predictor, mediator or outcome? How many levels does the categorical variable have?
Generally speaking, SEM software like lavaan can handle continuous, binary or ordinal variable, either exo- or endogeneous. For pure categorical variables, it is better to break it down into multiple dichotomous variables. 
If you don't like writing syntax like in R, there are implementations of mediation, SEM and path analysis in some of the libraries in Jamovi. A mediation option is also available in JASP if you prefer. Both software are free and easy to use. 

I agree with everything David Joubert said. Alternative software is Mplus (probably the most versatile) and Stata that can do quite a few of the models you may want to consider including path analysis and full SEM . I don't know if you are interested but Mplus is also good for latent growth modeling.