Audio engineer here. Definitely don't get a directional mic like a shotgun (fuzzy) mic or a cardioid mic (like singers use on stage). These require at least an audio interface, careful placement and multiple mics. For your situation, something like a Zoom H2n will be the best and perhaps cheapest option. It has 5 built in mics, so you'll be able to capture sounds from all over the lecture hall. Use the 4 channel surround option, or the X/Y or Mid-Side options, and see what works best. It won't be perfect, but it will be the easiest option to improve audio and simplify recording. Plus, the batteries last for many hours (ideally use the best rechargeable Eneloop Pro AA batteries).
The professional way to do it would require a mixing desk, an expensive DPA, Shure or Sennheisser lavalier mic on the lecturer, at least 2 room mics and possibly a boom operator with a shotgun mic to capture audience questions. (Although this last option could be replaced with room mics or additional fixed mics pointing at the audience). There are various options for room mics depending on the acoustics of the room, and there are no one size fits all solutions for professional quality audio.
But you don't need professional quality, just the best you can get. With a Zoom H2n, you can put it on a tripod near the lecturer, and it will pick up sound better than average human ears. It will be better on a tripod, because it won't pick up floor noise as much. To reduce floor noise further, get a decoupler to reduce handling noise and use a mini tripod sitting on a desk or chair. If the lecturer hall is windy (lots of air conditioning or windows and doors open) use a windshield.
You could use any handheld recorder that does X/Y recording and it might be fine. But 4 channel recording might help pick up more from the audience, and you can still boost the lecturer's audio by keeping the device close to them and having the "mid" mic pointing at them. The recordings can be fiddly to edit at first (especially if you want to boost part of them, like the "mid" or "sides", but it doesn't take long to get the hang of it. Use Audacity to edit the audio or any free audio editor you happen to have access to.
Please feel free to get in touch if you have any questions, and good luck with it!