What part of a dataset do I apply a traditional, statistical analysis to linear regression? Note: I've edited my question as recommended below by @EdM.
Suppose I have a supervised learning problem on a sizeable tidy dataset with real values—-e.g., the dataset has 100,000 rows or observations. I with to apply a linear regression and understand that there are two basic forms of linear regression analysis: explanation and prediction. If I desire the former, I study the relationships between the features and target variable of the linear model: some examples are captured by hypothesis tests, R-squared and F-statistics, and so forth. If I desire the latter, I use machine learning, by which I split the dataset into training, validation, and testing sets on which to fit the linear model and optimize it. By optimization or "learning" of the linear model, I mean the minimization of the model's mean-squared error by application of gradient descent and regularization.
My question is: suppose I wish to do both forms of analysis. Do I first do explanatory anlysis on the relationships between the features and target variables on the whole dataset, before I split the dataset and do the predicitive analysis, which imvolves the machine learning I described above?
 A: 
there are two basic forms of linear regression analysis: explanation and prediction.

That's too sharp a line, particularly if you're doing linear regression. Frank Harrell's course notes and book outline considerations for prediction, effect-estimation, and hypothesis-testing regression models in Section 4.12. They really aren't that different, except for the relevance of parsimony and the importance of model validation.
If you have a pre-specified model for your data, inference would best be done on the complete data set, as @DemetriPananos notes here. If your modeling process involves hypothesis generation or variable selection, however, then you need to involve some type of data splitting, as he explains.
For example, with a large data set you might be considering multi-level interactions among very large numbers of predictors and spline or GAM modeling of nonlinear contributions from continuous predictors, so there is a danger of overfitting. You might further be evaluating many such predictor combinations and choosing among them. Thus you have to take steps to avoid overfitting and to document that you didn't overfit.
In such cases you need some type of split-sample approach, using a portion of the data for developing hypotheses about the variables to select (including interactions and nonlinear modeling of continuous predictors), and then doing inference on the held-out test set, as @DemetriPananos suggests. If you can automate all of the modeling steps, the optimism bootstrap can let you build the model on all of the data and validate the model-building process if not the model itself.

If I desire [explanation], I study the relationships between the features and target variable of the linear model: some examples are captured by hypothesis tests, R-squared and F-statistics, and so forth.

An important point of statistical tests is to gauge how well your model might apply to the underlying population. With a very large data set, you could consider the held-out test set as your representation of the underlying population. Then you could develop your model with the train/validation subsets and evaluate overall measures like $R^2$, discrimination and calibration on the test set. (With a very large data set, hypothesis tests will tend to be "highly significant" even for effects that are small in magnitude.)
With a very large data set, each of the training, validation and test subsets are presumably large samples from the same underlying population. If your combination of using the training and validation subsets provided a useful and non-overfitted model, then the application of the model to the test subset should then show excellent calibration. In that case the distinction between "explanation" and "prediction" should be minimal.
