A 50 ohm output impedance is nice- it shouldn't come up a lot, most input Z are >1K and you won't see it. It sounds like you have a good instructor. You have these Op-Amp equations in your head from class... Gain = RF/RI but there are limits, its good to see this in action.
I have one old HP audio generator that has a 600 ohm output and it trips me up now and then. If you're in the return window, you might return and get something with 50 ohms output, its bit more versatile. Test equipment tends to hang around once you own it.
General purpose function generators don't have great level control, especially cheap ones, usually they use a pot and 10 x step attenuator before the final power amp. This can make for a little signal going into the final amp that tends to be noisy. Better generators generate a nice 1V or more leveled waveform, go through the power amp and then put a 50 ohm attenuator after the power amp. They are precision attenuating a clean signal and get a clean output. Function generators can usually add DC offset to outputs that can make this difficult, you don't want to put a 5v DC offset into a 50 ohms attenuator so they have a two stage output- AC path and DC path. Adds complexity= expense.
As far as brands and models, Uni-T is a lower tier supplier compared to Siglent. Siglent is kind of mid Tier but generally great value- some of the high end stuff is top tier- they make scopes that are sold by LeCroy that are first class (I own one). It all depends on what you want to do, its hard to know. There are so many goodies to buy, its hard to choose. After you buy all the equipment, buying cables and adapters is another expense and nice interconnects make lab time much easier. I do mostly medium frequency analog to low frequency RF. Everything needs a microcontroller so I have tools for that too- mainly a PC and some programmers.
I looked up the Uni-T you mention- here is a link to the manual I found-
https://res.cloudinary.com/iwh/image/upload/q_auto,g_center/assets/1/7/UNI-T_UTG900E_manual.pdfIt does have a 50 ohm output and says it can do a 10V signal into 50 ohm which is very good. Any generator will have a hard time driving a 5 ohm load when its not setup properly. There is something in the Utility menu that lets you select the output Z as 50, 75 or high Z- I suspect you have it set at the latter. The size is really nice and I think the quality is quite good really. Take a long bath and read the manual cover to cover. I think its a very capable instrument. I really like the 2 channels. This can be very nice for making modulated signals, exotic quadrature signals, etc. Distortion is good. freq counter input is nice feature. Great learning tool. 3 year warranty is great. Good manual. I think its a keeper.
When you're using a signal generator, think about what you're asking it do. If you really want to drive 10V into a 50 ohm load, you will have power peaks (V^2/R) of 2W- RMS is 500 mW. This is a sig gen not an audio amp. I use to stock a lab in a physics lab as a job in college. I also repaired equipment. My boss put internal fuses on the outputs of all the equipment and similar things on inputs. We spent a lot of time repairing equipment.
Edit- I notice there are some good reviews and discussion of your UNI-T here on EEVblog, search, conclusion thumbs up. One reviewer mentions that using a good USB power input is important and makes a difference.
Good luck in your schooling.