First, days of the week are included via a coding scheme. The most popular is 'reference category' coding (typically called dummy coding). Lets imagine that your data are represented in a matrix, with your cases in rows and your variables in columns. In this scheme, if you had 7 categorical variables (e.g., for days of the week) you would add 6 new columns. You would pick one day as the reference category, generally the one that is thought of as the default. Often this is informed by theory, context, or the research question. I have no idea which would be best for days of the week, but it also doesn't really matter much, you could just pick any old one. Once you have the reference category, you could assign the others to your new 6 variables, then you simply indicate whether that variable obtains for each case. For example, say you pick Sunday as the reference category, your new columns/variables would be Monday-Saturday. Every observation that took place on a Monday would be indicated with a 1 in the Monday column, and a 0 elsewhere. The same would happen with observations on Tuesdays and so on. Note that no case can get a 1 in 2 or more columns, and that observations that took place on Sunday (the reference category) would have 0's in all of your new variables. There are many other coding schemes possible, and the link does a good job of introducing them. You can test to see if the day of the week matters by testing the nested model with all of the new 6 variables dropped vs. the full model with all 6 included. Note that you should not use the tests that are reported with standard output, as these are not independent and have intrinsic multiple comparison problems.

• An r-score that's close to 1 indicates that the value response variable can be almost completely determined by the values of the predictor variables. Clearly this would be a large effect, but it is not a-priori clear that this is 'good'--that is an entirely different and philosophically thorny issue.
• It is not clear what they mean by 'r', given that you are doing multiple regression (where r is not typically reported). 'r' is a measure of linear, bivariate association, that is, it applies to straight-line relationships between (only) 2 variables. It is possible to get an r-score between the predicted values from your model and the response values, however. In that case, you are using 2 variables (and if your model is appropriately specified, the relationship should be linear). This version is called the 'multiple r-score', but it's rarely discussed or reported by software.
• R-squared is simply the square of r (i.e., r*r); it is not the standard deviation. It will also tend towards 1 as the relationship becomes more determinitive, not 0. Thus, if you think an r close to 1 is 'good', you should think an R-squared close to 1 is 'good' also. However, you should know that the multiple r (and multiple R-squared) is highly biased in multiple regression. That is, the more predictors you add to your model, the higher these statistics will go, whether there is any relationship or not. Thus you should be cautious about interpreting them.
• Sometimes output will list t-statistics for the individual predictors and an F-statistic for the model as a whole, in order to determine 'significance'. These are random variables that are computable by statistical tests and that have a known distribution when the degrees of freedom are specified.
• By comparing the realized value (that is, the value you found) against the known distribution, you can determine the probability of finding a value as extreme or more extreme than yours if the null hypothesis is true. That probability is the p-value.
• The t-value is used when you are testing only one parameter, whereas the F-value can be used in testing multiple parameters (e.g., as I discussed above regarding days of the week). The p-value associated with the F is the probability that at least 1 parameter is 'significant'. Another way to think about it is, 'does the model will all the parameters tested by the F included do a better job of predicting the response than the null model'.
• I am guessing that what you call the 'significance F' is the F-value that would need to be matched or exceeded for a test to be 'significant', presumably at the .05 level.

First, days of the week are included via a coding scheme. The most popular is 'reference category' coding (typically called dummy coding). Lets imagine that your data are represented in a matrix, with your cases in rows and your variables in columns. In this scheme, if you had 7 categorical variables (e.g., for days of the week) you would add 6 new columns. You would pick one day as the reference category, generally the one that is thought of as the default. Often this is informed by theory, context, or the research question. I have no idea which would be best for days of the week, but it also doesn't really matter much, you could just pick any old one. Once you have the reference category, you could assign the others to your new 6 variables, then you simply indicate whether that variable obtains for each case. For example, say you pick Sunday as the reference category, your new columns  / variables would be Monday-Saturday. Every observation that took place on a Monday would be indicated with a $1$ in the Monday column, and a $0$ elsewhere. The same would happen with observations on Tuesdays and so on. Note that no case can get a $1$ in 2 or more columns, and that observations that took place on Sunday (the reference category) would have $0$'s in all of your new variables. There are many other coding schemes possible, and the link does a good job of introducing them. You can test to see if the day of the week matters by testing the nested model with all of the new 6 variables dropped vs. the full model with all 6 included. Note that you should not use the tests that are reported with standard output, as these are not independent and have intrinsic multiple comparison problems.

• An $r$-score that's close to $1$ indicates that the value response variable can be almost completely determined by the values of the predictor variables. Clearly this would be a large effect, but it is not a-priori clear that this is 'good'--that is an entirely different and philosophically thorny issue.
• It is not clear what they mean by '$r$', given that you are doing multiple regression (where $r$ is not typically reported). '$r$' is a measure of linear, bivariate association, that is, it applies to straight-line relationships between (only) 2 variables. It is possible to get an $r$-score between the predicted values from your model and the response values, however. In that case, you are using 2 variables (and if your model is appropriately specified, the relationship should be linear). This version is called the 'multiple $r$-score', but it's rarely discussed or reported by software.
• R-squared is simply the square of $r$ (i.e., $r\times r$); it is not the standard deviation. It will also tend towards $1$ as the relationship becomes more determinitive, not $0$. Thus, if you think an $r$ close to $1$ is 'good', you should think an $R^2$ close to $1$ is 'good' also. However, you should know that the multiple $r$ (and multiple $R^2$) is highly biased in multiple regression. That is, the more predictors you add to your model, the higher these statistics will go, whether there is any relationship or not. Thus you should be cautious about interpreting them.
• Sometimes output will list $t$-statistics for the individual predictors and an $F$-statistic for the model as a whole, in order to determine 'significance'. These are random variables that are computable by statistical tests and that have a known distribution when the degrees of freedom are specified.
• By comparing the realized value (that is, the value you found) against the known distribution, you can determine the probability of finding a value as extreme or more extreme than yours if the null hypothesis is true. That probability is the $p$-value.
• The $t$-value is used when you are testing only one parameter, whereas the $F$-value can be used in testing multiple parameters (e.g., as I discussed above regarding days of the week). The $p$-value associated with the $F$ is the probability that at least $1$ parameter is 'significant'. Another way to think about it is, 'does the model with all the parameters tested by the $F$ included do a better job of predicting the response than the null model'.
• I am guessing that what you call the 'significance $F$' is the $F$-value that would need to be matched or exceeded for a test to be 'significant', presumably at the .05 level.