Lots of people use the RTLSDR dongles for budget radio astronomy.
One person, Marcus Leech, has led the way on this. His web pages should be consulted for any project in this area.
One doesn't need a fancier SDR, from what I understand, an RTLSDR is a unique value and works quite well even compared to much more expensive SDRs at the hydrogen line frequency. Also, the cheapness and small size offers big placement advantages. (you can put the receiver near the antenna without worrying as much about theft of an expensive device left outdoors, perhaps feeding it with a USB extension cable instead of coax)
For a student project especially (probably thousands of students have done this as part of their education) it offers a lot in terms of software value that you lose by using less well supported SDRs. (make sure whatever SDR you use is supported well by Gnuradio libraries, many but not all are.)
Combined with a modern LNA made with a decent low noise MMIC like the Minicircuits PSA4-5043+ - which will give you a good low noise figure well under 1 db, you should do fine. The biggest drawback I can see is the lack of dynamic range and some particularities of how the dongles handle signals and noise (see sm5bsz.com for an examination of this with different receivers in detail) there are AGC peculiarities inherent to using a device optimized for DTV, not radio use and certainly not radio astronomy!
One mid size dongle that usually comes in white or black can be recognized by its elliptical curve design of ventilation holes on one side.
It uses a flat SMD crystal which has all four corners of the PCB at ground potential so its easy to detach from the USB connector and case and turn into a part of a larger PCB.
They cost around $7-10 on the usual auction sites.
It has some advantages over many of the other cheap dongles for a hardware project. Its more stable than the other cheapies, there is less drift, and the initial error is always lower. Spend a bit more and you'll get a TCXO but it wont be an adjustable one.
There is no need to use the RTLSDR blog version #3 for example for this kind of project. (although thats the one I would buy if I was only going to get one, and use it indoors) They cost around $25 and represent the current state of the art in rtlsdr dongles, because they have responded to lots of requests from the user community and really taken the platform as far as it could go.
Antenna?
The simplest and cheapest antenna with enough gain to get decent resolution would be a horn antenna. Not a dish which at 1.4 GHz is not going to give you as much gain as people routinely get at much higher frequencies. Much easier to manage cost wise.. MUCH easier to use for the duration of a science project, and less obtrusive than a dish. You could use a large cardboard box like the kind refrigerators come in as its enclosure and support, and a wave guide - ideally a horn covered with aluminum foil gently sloping towards its end, inside as the waveguide (but the horn needs to be angled at an optimal value and structurally stable so use foam core to make it). At its bottom, receiver end you need a probe of the right length at the right dimensions especially distance from the far end, for the frequency you want to receive. (also make sure that any rain water that is inadvertently collected by the horn has a place to egress that doesn't allow your LNA or feed cable to get wet! The LNA must be well waterproofed or it may start to oscillate causing problems of all kinds for you and others.)
You can use foam core coated with aluminum foil for your horn - use some kind of spray on adhesive thats waterproof if you can find one, and aluminized mylar duct tape (which is extremely strong) to make the horn.
If you need to rotate the horn for any reason (tracking a satellite, for example) you could perhaps mount two flat wooden boards and a swivel on the bottom of the box to allow it to be pivoted and something similar on the sides (bungee cords might do) to let the horn inside to be tilted in the vertical plane?? Having protractor like markings to show the vertical angle and allow the base to be calibrated with true north as "0" would also be useful.
Shield the dongle in some kind of fully conductive enclosure and use multiple ferrite beads on the USB cable (thats really crucial) (so its internal noises don't find their way into the receiving system especially ahead of the LNA.) An additional polarized cap of sufficient value to filter the DC power a bit better, but not so large that it prevents successful USB insertion due to a DC inrush thats too high, is ideal.
Here's what you should do. If you use an LNA, either feed it via its feedline from a good bias tee or from external DC wire but- if you do that, you must both enclose it very well (shield it so that the LNA is not exposed to RF via any other than its input, especially not via the power line route) and use a high quality feed through capacitor of at least 1 nf meant for use as an inline RFI filter. And this is most important, make sure the power line is presented to the LNA via a non-resonant method, which is tricky, consult bias-tee references... also after the external DC line exits it should go right along the feedline and not hang down or away from the support, and not present an attack surface for external RF to use to get into the LNA.
Whether you use the coax for power or DC, the input to the LNA should not go straight to the antenna. There should be some fairly short length of the highest quality coax between the LNA and the feed antenna, but not a super lot. (depending on your installation, but no more than a meter and if you use that much you should use a very high quality grade coaxial cable or semi-rigid line of no more than the minimum length but more than say 10 cm. And you should run that portion of the feed line right along any metal supports to the actual feed. Ive just found this to be a critical factor when using this kind of LNA for best performance.
And also make sure the PCB of the LNA has been cleaned really well of flux. Squeaky clean.
The LNA should be constructed as simply as possible. Just one high quality ceramic dc block capacitor the MMIC and the output+power connection. It should be right at the antenna and it and all connectors weatherproofed.
Instead of coax, to the dongle from computer (indoors?) use a powered USB extension well equipped with lots of ferrite split beads to suppress RFI.