They're not truly invisible
Fundamentally, if something cannot be detected by any form of electromagnetic radiation (aka Light), then it cannot interact with matter. Physical contact, chemical reactions, lasers, all these things are based around the same fundamental force, electromagnetism. If your aliens can't be detected with electromagnetic radiation, they can't interact with the Earth at all; if they can interact, they can be detected.
How are they reducing their signature?
So our invaders are classically invisible: They can't be detected with visible light. Per the question, they also can't be detected in the Xray or IR bands. There's only a few ways they could do this.
These are already being developed today around the world. Essentially metamaterials allow lightwaves to break around them, then reconstitute on the far side; by the time they reach a detector, there's little sign of the change.
These materials, however, are very limited in the wavelengths they can disrupt, so the aliens would be visible to detectors working in the radio, microwave, UV, or gamma wavelengths. Fortunately, our radar works at just those wavelengths (the question mentioned that these ships were detected on radar). The fix is easy, then - use radar. Nearly all weapons are already radar-controlled, so if the baddies are expecting us not to shoot at them just because we can't see them with our squishy organic eyes they're out of luck.
Essentially, rather than trying to bend light around themselves, they're generating appropriate images and projecting them onto the outside of their hull. That is, if you look up at their ship, you're actually looking at a picture of the sky and clouds that they're projecting onto the bottom of the ship.
This works reasonably well - from one angle. If you and your friend both observe the ship from a mile or so apart, one or the other is going to see a discontinuity, and detect the ship. Also, this doesn't work against active detectors (like radar).
They're folding space
This is one of the weird ones. Rather than being classically invisible, these aliens are folding space around their ships so that they're always inside a pocket of spacetime that doesn't interact with the rest of the universe. They get themselves into position, then open a narrow window to shoot through.
This is true invisibility. Unfortunately, it doesn't fit with radar being able to pick them up. It also prevents them from seeing anything that's happening outside before they open their gun port, so if you can predict what they're going to do, you can have an entire army waiting outside when they open up (or, y'know...a nuke).
Dunno how they do it. It's not important. They're invisible, okay?
Okay. Not much we can say about that.
Okay, so how do we detect these guys?
Your question already gave the answer - radar. Radar is a very common tool; almost every part of the developed world is covered by radar, and most militaries have some form of mobile radar systems available, either ground based, ship based, or airborne. Basic portable radar systems could be easily adapted from police speed detectors for infantry use.
Secondly, almost any form of invisibility - except perhaps the pocket-universe one - is going to have one great big problem: heat.
Your alien ships are going to have to vent waste heat or they'll cook themselves That heat bloom is going to be detectable by IR sensors, even if nothing else is; follow it to the hottest point and stick a couple of nukes up their back end; aliens go bye-bye.