@Ondrej and @Michelle have provided some good information here. I wonder if I can contribute by addressing some points not mentioned elsewhere. I wouldn't beat yourself up about not being able to glean much from the data in tabular form, tables are generally not a very good way to present information (cf., Gelman et al., Turning Tables into Graphs). On the other hand, asking for a tool that will automatically generate all of the right graphs to help you explore a new data set is almost like asking for a tool that will do your thinking for you. (Don't take that the wrong way, I recognize your question makes clear that you aren't going that far; I just mean that there will never really be such a tool.) A nice discussion that is related to this can be found here.
These things having been said, I wanted to talk a little about the kinds of plots that you might want to use to explore your data. The plots listed in the question would be a good start, but we might be able to optimize that a little. To start with, making "a large number of plots" correlating pairs of variables might not be ideal. A scatterplot only displays the marginal relationship between two variables. Important relationships can often be hidden in some combination of multiple variables. So the first way to beef up this approach is to make a scatterplot matrix that displays all pairwise scatterplots simultaneously. Scatterplot matrices can be enhanced in various ways: E.g., they can be combined with univariate kernel density plots of each variable's distribution, different markers / colors can be used to plot different groups, and possible nonlinear relationships can be assessed by overlaying a loess fit. The
scatterplot.matrix function in the car package in R can do all of these things nicely (an example can be seen halfway down the page linked above).
However, while scatterplot matrices are a good start, they are still only displaying the marginal projections. There are a few ways to try to move beyond this. One is to explore 3-dimensional plots using the rgl package in R. Another approach is to use conditional plots; coplots can help with relationships amongst 3 or 4 variables simultaneously. An especially useful approach is to use a scatterplot matrix interactively (albeit, this will require more effort to learn), e.g. by 'brushing'. Brushing allows you to highlight a point or points in one frame of a matrix and those points will simultaneously be highlighted in all of the other frames. By moving the brush around, you can see how all of the variables change together. UPDATE: Another possibility that I had forgotten to mention is to use a parallel coordinates plot. This has a disadvantage in not making your response variable distinct, but could be useful, for example, in examining inter-correlations amongst your X variables.
I also want to commend you for examining your data sorted by date collected. Although data are always gathered over time, people don't always do this. Plotting a line graph is nice, but I would suggest you supplement that with graphs of autocorrelations and partial autocorrelations. In R, the functions for these are
I recognize that all of this doesn't quite answer your question in the sense of giving you a tool that will make all the plots for you automatically, but one implication is that you wouldn't actually have to make as many plots as you fear, e.g., a scatterplot matrix is just one line of code. In addition, in R, it should be possible to write a function / some reusable code for yourself that would partly automate some of this (e.g., I can imagine a function that takes in a list of variables and a date-ordering, sorts them, pops up a new window for each with line, acf, and pacf plots).