How to Analyze Continuous IV and DV?

#1
I'm a relative novice at statistics and would appreciate some help with a dataset I'm trying to crunch. I'm currently looking at factors that predict increasing utilization of healthcare resources following surgery in a set of 130+ patients. Some of the independent variables I'm looking at include age, BMI, Marital Status, Length of Stay after Surgery, Smoker Status (Y/N), etc. The dependent variables are number of ED visits, readmissions, post-op visits following surgery.

Initially, I was planning to run univariate linear regression on the data set, followed by multivariate regression of the significant predictors. However, I tried running the analysis with age vs. ED visits on SPSS and got the following output after checking for assumptions of linearity and normality of residuals:

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So based on this, I feel like linear regression is not the way to go and I have to consider analyzing the data via another method. Currently, the variables are coded as continuous variables (both age and ED_visits). What are your recommendations on how to run the data, or maybe recommendations to transform the data in some way to fix better meet assumptions of OLS? Other thoughts I've had include transforming the dependents into nominal/ordinal variables and doing binary/multinomial logistic regression or doing a Poisson regression, but honestly not sure if either of these would solve my problem. Thanks for the help!