The Moon is very difficult to photograph with much else in the shot as it's so bright. Most night pics with the Moon and Stars, or the Moon and Landscape are two shots combined, a fast shutter speed shot for the Moon and a brighter exposure for everything else.
Also the Moon is smaller than most people think, so looks small in any even slightly wide-angle shots. Using a long lens and shooting from a LONG way away can help, but again most photos with large Moons are composites and the Moon probably was shot on a different occasion with a longer lens. (One experiment to get a feel for its size - I tell people to stretch their arm-of-choice out full-length in front of them and judge how much of their hand would be necessary to completely hide a Full Moon at that distance (guess then see the P.P.P.S.).
You can rent camera gear, but then you have a limited time to learn how to use it. (In the U.S. I understand LensRentals is good.)
You could shoot the Moon with your bridge camera and use something else for the Landscape, but then needs editing.
Many cameras do have HDR (High Dynamic Range) modes where they shoot multiple shots at different exposures and combine them in the camera. Taking multiple shots at different exposures is called bracketing. To have more control you can use separate HDR software to combine images later (either from bracketing or if the camera's HDR mode allows you to save the source images), rather than having to use a full photo editing program like Photoshop. If you can find the old, free, Google version of Nik Tools online (shouldn't be difficult, you just need the version number, which is 2.2.24.12) you could use HDR Efex Pro 2 (I'll add a P.S.). I like Aurora HDR followed by Franzis HDR Projects Pro, the former is often given away as an old version for things like photo magazine subscriptions, the latter often sold at large discounts or old versions given away on sites like giveawayoftheday. Note ideally the images need to be aligned for HDR to work, you need more complicated tools to import and merge a separately shot Moon (Photoshop, Afinity Photo, etc.).
As to the camera and lens, any digital camera made in the last few years will be fine, you just need to use a tripod and keep the ISO down, and hence use a longer exposure and a tripod or something. Where modern camera sensors really gain is in noise performance at higher ISOs, but providing nothing is moving (so ignore windy days) you can just take a longer shot. Ditto for the lens, although a faster (smaller number after the f/) one will allow a shorter exposure and help with any movement. However the wider the aperture the narrower the range of stuff that is in focus, although that also increases with a wider lens.
Suggestion. Try using a tripod with your bridge camera, shoot at whatever is its Base ISO (often ISO 100, might be 200), take a long exposure and see what you think, for the Moon add a short one and combine the two. If you want more pixels and have a PC/Mac try Topaz Gigapixel AI at 4x on it (or send me the image and I will). If that isn't good enough a 24-30MP APS/FF camera with a 24, 35, 40 or 50mm f/2.8 or less lens should be fine (20mm is cool for very wide, the Moon will be teeny tho), that is unless you want to print huge and look at very close.
Bonus - here's my guide to using stacking software to make better quality Moon images, when just photographing the Moon alone:
The Moon, with its colours enhanced... - posted in Major & Minor Planetary Imaging: So I decided to have a go at Astrophotography with my AP Stowaway. I dont have the field flattener, but did spend nearly 7 quid on a 2 to T2 adaptor and a T2 to Canon EOS Adaptor. I went out last night and took...
www.cloudynights.com
P.S. Using HDR Efex Pro 2:
Note the Nik tool is a Photoshop Plugin but you can just (on Windows) drag a link to the HDR Efex 2.exe to the desktop and go from there (although IIRC it just saves the result in a "standard" location rather than offering a choice).
If you have Lightroom that also has built-in HDR support.
Here's an example of an Aurora HDR giveaway for AP subscribers:
In partnership with leading software developer Skylum, we are delighted to offer AP readers the chance to get their hands on the award-winning Aurora HDR 2018 software for free. All you need to do is to enter your email address. Aurora HDR 2018 is available for both Mac and Windows and features...
www.amateurphotographer.co.uk
Ditto for HDR Projects (but expired, GAotDs are just for one day), note comment #1 is me explaining how to use it:
Note: Franzis changed the title of the product. HDR projects 6 was formerly known as HDR projects 2018. HDR projects 6 transforms your photos into Ultra-realistic...
www.giveawayoftheday.com
P.P.S. HDR:
You can have a very wide range of brightness in a scene, more than any camera sensor can cope with. A very (!) bright scene may have 20 stops of Dynamic Range (the range from the lightest bits to darkest bits, a stop is a doubling or halving of the amount of light captured, so 1/60th to 1/125th or f/2.8 to f/4, etc.). In such a scene whatever exposure you choose a lot of the detail won't appear as the sensor sees the darkest stuff as pure black and/or the brightest stuff as pure white (you should get everything on a dull day with 8 stops of Dynamic Range in the scene though). However by taking 3 to 5 (say) 12-14 stop images (the range a large sensor may cope with) separated by (e.g.) 3 stops you get information for 20+ stops of brightness (so for the Sunny Window in the dark room you have a shot showing what is outside, but that shot won't show what's inside, it will be mostly black; but you have another shot with the interior detail and the window is white). The HDR software then chooses to use the bright detail from the first image and the dark detail from another. It must map that range onto the 8 or 10 bits of brightness your monitor supports (or whatever your printer allows, etc.), but you still see a lot more Dynamic Range from the original scene. One issue is making it look natural, although some people like more unnatural looks, that's for the photographer and audience to debate (personally I'm not a fan of overblown HDR effects except in rare subjects).
To clarify shooting at +/- two stops means with a shutter speed 4x and 1/4x the correct exposure. Don't adjust the aperture as that will change what is in focus. For three stops it's 8x and 1/8th. So for example three images at +/- 3 stops could be 1/60th, 1/500th, 1/4000th or five images at +/- 2 stops: 1/30th, 1/125th, 1/500th, 1/2000, 1/8000 (where the centre of the group is the correct exposure for most of the scene). Often three images at +/- 2 stops is fine, depends how much detail is very bright or very dark in the image. It's best to use a Tripod (or random bit of scenery to rest the camera on) to shoot the bracketed images, but if they are shot close together (many cameras have bracketing options that will shoot the images in a short burst, ideally with stabilisation in-lens, in-body or both), or even HDR options where you can choose to keep the original images as well as the HDR image made in-camera, then you can still get good results. Most HDRs I've done were shot hand-held but for a night scene a tripod (or just resting the camera on something, bean-bags are good) would be necessary. Also note camera DR decreases about 2x every time you turn the ISO up, so shoot at the lowest ISO possible (ignoring special low ones you have to enable separately).
P.P.P.S. With your arm fully stretched out the Moon will be covered by the tip of your little finger. (Many people guess it's their fist.)
