Generally, you can't tell anything just from the statistics and that it is because statistics should be applied only after you fully understand the problem at hand. What is the underlying context of that problem? What are the questions that are important to answer? How will the answers be used? Who will need to digest those answers to act on them and what is their level of statistical sophistication? All of this drives what type of statistical methodology will be applied, how it will be applied and how the results produced will be presented and disseminated.
The famous Anscombe dataset quartet was created specifically to illustrate the perils of using just the statistics to conduct a regression analysis. This quartet includes 4 distinct datasets which include a variable y (response) and a variable x (predictor). If you don't visualize the data in each dataset and just blindly compute the R-squared value for each data set from a simple linear regression of y on x, you would find that the 4 datasets produce identical R-squared values. However, if you take the time to construct a scatterplot of y versus x for each dataset, you would see that a simple linear regression will not even make sense in some cases. See https://data.princeton.edu/wws509/stata/anscombe, for example.
As @passerby51 indicated in their answer, when it comes to modelling nonlinear relationships, you have two different routes available to you:
- Assume the nonlinear relationship is parametric;
- Assume the nonlinear relationship is non-parametric.
Parametric nonlinear relationships - of which polynomial regression of order 2 or higher is one example - can be described by a relatively small number of unknown parameters which must be estimated from the data. See this blog post for a nice overview of possibilities when it comes to parametric nonlinear relationships: https://www.statforbiology.com/articles/usefulequations/.
Nonparametric nonlinear relationships allow for more flexibility since you let the data speak for themselves in determining the shape of the relationship - you don't impose your preconceived ideas on the nature of that relationship (e.g., I think the data should follow a quadratic relationship).