There is no Imagen website, app or download. Go to gemini.google.com, or install the Gemini app, and sign in with a free Google Account — that is the entire setup. Searching for "Imagen" lands people in Google Cloud's developer console, which opens by asking about projects and billing, or on a lookalike site with nothing to do with Google.
You can make pictures without paying. Google's plan comparison lists "image generation and editing" on the free plan, alongside several paid plans. Its two plan pages disagree — the Google One version lists image access only on the paid rows — so assume the free allowance is real but modest.
The clearest free-versus-paid line is the download: Google's help page says images download at 1K resolution without a Google AI plan and 2K with one. 1K is fine for a screen or a slide; small once you want to print or crop in hard.
The limits are not counted in pictures. Google describes usage as compute-based across everything you do in the app, refreshing every five hours until a weekly ceiling — so a long chat earlier in the day eats into the images you were saving. Subscribers who hit it drop to a lighter, faster model; Google does not say what a free account gets there, so treat running out as "come back in a few hours".
Google's help page puts generating images at 13 and over, or the applicable age in your country, and editing at 18 and over. Availability follows the Gemini app's supported countries and languages, and school accounts may be restricted further.