Are assumptions for multiple linear regression basically the same as simple linear regression? One has to check for linearity for each of the continuous predictors versus the outcome variable? If there are categorical variables, one has to check the linearity between each of the continuous predictors with the outcome variable for all levels of the categorical variables?
The assumptions are about the same: linearity, reliability, homoscedasticity, normal distribution, error independence and normality, no outliers, no collinearity, and so on.
The primary special issue with multiple regression is multicollinearity. That requires some testing distinctive to multiple regression - tolerance is one method and variance inflation is another. If you want more details on how to do this, don't hesitate to ask.
One generally examines the conformity with regression assumptions by examining the residuals since most of the regression assumptions have to do with the distribution of residuals rather than with the distribution of the outcome variable or covariates. In models with interactions or spline terms, the analysis may be more complex, but basically, the answer is yes, you would be checking the homoschedasticity and presence of autocorrelation and lack of linearity within each covariate.