I am a member of Astropeiler Stockert e.V., and we are fortunate enough to be able to approach this problem coming from the "large side" :-) We have a 25m, 10m and 3m telescopes as well as an interferometer made from two 1m satellite dishes available. All these dishes can be used to do interesting things, but you'll need to match the instrument to your target (and, in a hobbyist setting, often also the target to your instrument).
First of all, you should think about what frequency band you want to work on. 21cm (1420 MHz) is the classical hydrogen line, which lends itself well for mapping neutral hydrogen in the spiral arms of our galaxy. You can expect comparatively strong signals in a quiet frequency band there. Lower frequencies (and large bandwidths) are interesting for pulsars, higher frequencies give access to more interesting phenomena but will require a lot of work on the high frequency side. So my recommendation would be to start out with 21cm.
Secondly, what to look for? In general, the following areas of observation are easily accessible for amateurs:
- continuum emission of the Milky Way, perhaps up to creating maps
- spectral measurements of selected, "bright" (as in "intense radio sources") objects
Let's look at them in detail:
Continuum measurements are easily done using dishes of any size, but I'd start from 3m upwards to get more interesting results. Calibration of the backend receiver is not trivial, and you should plan a bit of time for it.
Spectral measurements also benefit from larger collecting areas, but you'll also need to set up a suitable backend. They are, however, possible from 3m upwards if you are happy with a bit of mapping of intra-galactic velocities.
Pulsars require a lot of collecting area and bandwidth, so they are mostly the domain of large instruments. We are currently observing several dozens of the brightest pulsars on our 25m dish. The 10m might be sufficient for some of the very brightest. And while there is a report of pulsar measurements using a 3m dish and an RTLSDR frontend, this is a feat that requires experience and dedication. So, I'd recommend this only for 8m and above dishes.
Interferometry with 2x1m dishes (20 GHz) is rather interesting, but very involved in terms of analysis. Two receiving systems help mitigate local fluctuations, and with that setup, we can observe sources down to 2 Jy (with a looong integration time). This setup will give you access to one or two handful of interesting targets like M1, W51 or Cyg A.
Finally, I'd like to recommend the EUCARA conference series (European Conference on Amateur Radio Astronomy) and the SARA group as great starting points. They have conference presentations available online that show what other amateurs are doing.