Imagine a sci-fi scenario where there is, in effect, an arms race between countries in terms of investing in SETI-style efforts, each country wanting to make sure that if there is any intelligent life out there to find, they will be the first to find it. A country like the US invests a trillion dollars in the effort - what might this trillion buy them? Would there be economies of scale to be realized, or would there instead be huge diminishing returns?
They could start imaging exoplanets. The FOCAL telescope is a proposed design where a telescope is sent to a distance of at least 550 AU (82 billion km) and uses the gravitational lens of the sun to image an exoplanet enough to get detailed images of the surface including continents, oceans, determine levels of atmospheric gases and even see if there are cities that light up at night like we have on earth.
The downside is that each telescope can only look in one direction, so NASA would have to send fleets of these telescopes. The furthest we've ever done so far is Voyager 1, which is only at 155 AU after 44 years of travel, so it will require new methods of spaceflight to get there. This video explains the idea of using solar sails, so a spacecraft could buzz close to the sun and then get blown out to 550 AU very quickly: https://www.youtube.com/watch?v=NQFqDKRAROI
Side Note: If they wanted to test this idea, they don't have to spend a trillion and wait a decades to get out to 550 AU. There's an idea for a Terrascope which uses the Earth's gravitational field and a much closer detector.
Considering that the Apollo program cost around $300bn adjusted for inflation, a trillion dollars being spent would be quite the sight to behold. It's also so much money, that even spread out over a long time-span, there will be difficulties spending it. Because of that, a multi-pronged approach would be likely, where any half-decent and remotely feasible idea gets basically infinite funding.
Here are some of those ideas:
Space is full of transmissions we just don't understand. Due to our limited understanding of encryption and radio technology, what we interpret as "white noise" using our radio telescopes is actually internet traffic that carries alien memes or whatever. To attempt decryption, we build a vast supercomputer and create an AI to analyze this noise and search for patterns.
There is life on Mars/in our solar system. The current scientific consensus agrees that there was liquid water on Mars, and most think there still is to some degree (in slush or underground lakes). The idea that microbe-analogue life exists on Mars right now has a decent chance of being true. With this huge infusion of money, and the advanced technologies available to us today, unmanned missions to all solar bodies will become near routine, and manned missions will likely stretch past the asteroid belt to investigate Jovian moons for signs of life.
We just haven't looked close enough yet. When Hubble started sending back good science data, it revolutionized physics in many ways--all from only having a mirror less than three meters in size. JWST will doubtlessly do the same with its 6.5m mirror. With a functionally infinite budget, we can start considering truly ridiculously big telescope concepts, including gravitational ones that use the sun as a lens or converting a crater on the far side of the Moon into a massive radio telescope. Really, imagination is the limit here.
Assuming life was planted in our solar system by external intelligences, these intelligences might have left messages for us. The rational place to leave such messages would be in the DNA as this would be replicated and so maintained, where as a note in a bottle will probably not persist 4 billion years. Cryptographic analyses of conserved DNA across a range of organisms with an eye towards reconstructing the basal state of life will look for a message beyond the function of that DNA.
But 4 billion years is a while. Maybe it would be good to look someplace that evolution and mutation had not been going on for so long. Mars was nice 4 billion years ago and would probably have been seeded with life too. Maybe Venus as well. Perhaps DNA in the life from those days might still be preserved there. Search for remains of that life will ensue, with the idea that the message as preserved in that DNA will be cleaner than what exists here.
I am trying to think of what sort of message I would send to my descendants 4 billion years in the future. I think it would be a joke.