Get the test. This time of year, as you suggest, their turn around time shouldn't be too long (you can call them and ask.) The info on the results should be fairly easy for you to make sense of.

As far as buying anything, well, if the turn around time is fast, and the plants haven't really grown large, and that "etc." in your comments includes soil, then what you have should support the growth at least until you get the results.

Not all compost is created equal. Some has sufficient nutrients, some - well, not so much. The peat basically has nothing. But, even a cheap soil should have enough nutrients for a week.

If it is a new bed that you've never used before, or especially tested before, get it tested ONCE by your university extension. Excellent investment. Don't waste your time with a self test. The reliability of those is suspect. That information from a professional test could be profoundly valuable, if some real deficiency is identified. You could match that deficiency with symptoms that you see in your plants, and amend using those symptoms as an indicator of your success in treating that deficiency. To the extent that test reveals things that you really can't treat well, like soil alkalinity for example, you at least know what you can't grow.

You might talk with your neighbors, who may have had the test done, and probably share the same type of soil.

If you feel like you need to fertilize, then sure, use a general purpose fertilizer. For many people, fertilization is a treatment for the psyche of the gardener more than it is for the plants. But go easy on it until you know the results from your soil test.