Model selection
Based on the information you provided, I assume that you have estimated all the models. If so, you are basically asking how to choose the best model. There are a couple of model selection criteria. The most usual methods are as follows:
The $R^2$. A high $R^2$ indicates a good fit. If you use different independent variables in each model, it is better to use the $\bar{R}^2$ (adjusted $R^2$). Note: don't use this metric to pick the best model if you want to forecast, there is a lot of risk of overfitting.
Information criteria such as the AIC and the BIC. These metrics are defined as a loss function plus a penalty term (AIC and BIC differ in the penalty term). A loss function should be minimized, so choose the model with the lowest AIC or BIC.
These slides also mention a Mallows $C_p$, but I have never heard of it during my Bachelor nor during my Master in econometrics.
There are many more methods to decide the best model, but are probably too advanced for beginners in econometrics.
To (hopefully) give you more intuition about the different estimators I suggest you read this guide. It may be a bit technical, but it lists all the advantages and disadvantages of the different estimators.
The Hausman test
The high P-value indicates that you should reject the $H_0$. The $H_0$ in the Hausman test is that the RE is preferred over the FE due to higher efficiency. So yes, you should use the FE estimator.