Composting fruit and vegetable peelings and cardboard (if you have outside space for compost bins, yes 'bins' plural) is very satisfying, a little bit magical AND reduces landfill.
Every week we put fruit and veg peelings plus shredded cardboard into our 2-compartment tumbling compost bin. We fill the compartments one side at a time. Once both compartment sides are 3/4 full we transfer the most rotted-down matter (from the side we had started filling first) to our wormery bags, for the final soil-making-magic. Composting = waste removed without the need for rubbish collectors or wheelie bins or landfill, plus free garden soil to fertilise the garden with. Double hurrah! The crumbly black 'soil' produced by successful composting is too rich to plant seedlings into directly, mix one trowel of home-made compost with a bucket of garden soil (i.e. a 'normal' orange B&Q 12 litre size bucket) and mix well before putting it anywhere near roots / seeds. Or spread the crumbly compost you make onto a dormant garden bed in the Autumn, ready for planting up in the Spring.
This is what goes into our compost bin...
Yes to - most fruit peelings (well chopped up), most veg peelings (again, well chopped), pumpkins which are past the eating stage can be chopped well and added, tea-leaves (loose or removed from the tea bags), coffee grinds, clean and baked and crushed egg shells (when not being used to deter slugs), sellotape-free and non-shiny cardboard (ripped up), egg boxes (ripped up) etc. In the picture above you can see that I have a large, 6 litre, lidded indoor collecting bowl in the garage (this or this container would work too), so I need only trek up the garden to 'feed' and tumble the compost once a week.
Pumpkins should be cut up into small pieces to ensure that the rind breaks down completely in the compost. Obviously remove any non-compostable decorations and don't compost painted pumpkins (unless it is non-toxic washable paint and you've washed most of it off), varnished pumpkins or pumpkins which have been previously soaked in bleach!?! Pumpkins are very moisture rich, so add plenty of dry shredded cardboard to compensate for this in order to keep your compost happy.
No to - onion skins and trimmings (very stinky when rotting), avocado / peach / nectarine stones (they take ages to rot), grapefruit peel (my mum told me not to add them and she has been successfully composting for years, before composting was even a 'thing'), meat, cooked fruit and veg (attracts flies and vermin), tea-bags (I empty the tea leaves out as most tea bags contain some plastic and most of those which don't aren't actually 'home compostable'), garden weeds, leaves and grass clippings (too much of them for a tumbling composter to manage, so they go into the council garden waste wheelie bin), rhubarb leaves - tearing them up for tumbling composter means you'd get regularly covered in their toxic oxalic acid juice, shiny and inked cardboard - I don't want to mix inks and dyes into my veggie patch...
You also need to add A LOT of cardboard (shredded or ripped up into 2cm squares if using a rotating composter) to the fruit and veg waste to get the right balance for successful compost-making. 2x or even 3x as much ripped-up cardboard as fresh peelings apparently. We try our best. Online-shopping-whilst being-at-home-more has certainly boosted our cardboard supply. Some shredded paper can be added, but paper gets soggy and clumps together (whereas the cardboard acts as a good separator in between soggy peelings). If you don't add enough dry 'brown' waste (i.e. cardboard, chopped up straw, dry leaves, a bit of slow-to-rot sawdust) then the compost will not manage full decomposing - leaving you with clumps of sticky brown almost-rotted matter. Don't fear if this happens, poke the clumps apart with a trowel and add lots more dry, torn up cardboard and mix well. Repeat this extra-care until the compost is no longer clumped. Then be patient and wait for the decomposing to get going again.
*Update* For just under £50 we got this 'Bonsaii' heavy-duty paper shredder (which can handle 12 sheets of paper at a time) and it happily shreds cardboard! Game changer. A few warnings - I wouldn't try to shred 'thick' cardboard (so Amazon-box cardboard is as robust as I would risk putting into the paper shredder). Be sure to remove all plastic tape and all the weird black Amazon paper-but-got-string-in-it tape, or you will gum up / jam the blades. Also, I always stick faithfully to the shredding time instructions, to avoid overheating the shredder (ours runs for 5 minutes and then needs to be switched off to 'rest' for 45 minutes before its next use). Ours has been used to shred cardboard for 5 minutes once a week for 6 months and is still going strong.
In the past we have had problems with rats being attracted to 'dalek' style compost bins and problems with fruit flies swarming. So that is why this time we chose an expensive 160 litre (2 x 80 litre chambers) sealed, rotating compost bin (like a big tombola you'd see at a traditional school fete) -see ours in the image below, or for an online image... ours is very similar to this one. No problems with vermin or flies or smells this time. But the composter is VERY HEAVY to rotate when both chambers are almost full (one with well-rotted matter and the other with fresher matter). At this stage I can barely rotate it, so I'd suggest choosing a different tumbling composter design - there is one with a handle on getcomposting.com (which is subsidised as the website is linked to local UK councils, although is still £100+).
By trial and error we found that in the UK climate a rotating composter won't usually produce fully rotted compost on its own. It's simply not warm enough for long enough. Rotating composters rely on bacteria and fungi action, but we needed more composting power, so we added an extra stage.... worms. DO NOT add worms into your rotating composter - the worms would be bashed around whenever you turned it and if that didn't kill them then they would fry as the sealed chambers get very hot in summer. It is advised to place your tumbler in direct sunlight, so it heats up to temperatures required to kill any weed seeds (so it is NOT a worm-friendly environment). We transfer the most rotted material from one side of the tumbler (once both sides are full) via a large garden trug placed under the tumbler, into 15 gallon green, zip-up, cylindrical garden bags. This leaves one side of the tumbler free to begin refilling and it leaves the other side of the tumbler alone to continue rotting down. Into the green zip-up garden bag of partially rotted material, we add worms (100g of 'Tiger' worms from Yorkshire Worms is perfect for one 15 gallon compost bag). It is VERY important to put the wormery bags in the shade so they don't overheat. We leave the worms alone to do their job in the compost bag, until the compost in the bag looks brown and crumbly and soil-like. Every six weeks or so I do like to empty the wormery bag out into a trug for a gentle mix, to ensure the worms don't just work on the very top layers (and so I can check on their progress). In the Autumn, we tip any 'ready' homemade compost onto a bare garden bed (quickly picking up as many of the red Tiger Worms as we can to use in other compost bags!) Then we slightly dig the patch of ground over, to mix in new and old soil (but we don't dig more than 3cm down as we are aiming for a 'no dig' approach to ensure a healthy soil structure). The compost bag (and worms) are then ready for the next batch of slightly-rotted-down tumbled material. As a household of four, we have FOUR compost 'worm' bags on the go, all at differing stages of compost production. The worms reproduce so well that we've only had to buy one 100g bag of worms so far!
Other options....
If you have a large enough garden then a pair of wooden-pallet compost heaps or two council-subsidised black 'dalek' compost bins (the ones we found attracted pests too close to the house in a small garden) are worth considering. I'd always aim to have at least a pair of compost bins (or a small two-compartment bin like ours plus extra 'storage' wormery containers) so that you can leave older compost to rot down fully whilst you continue to add fresh matter to a newer batch. If you choose a non-rotating compost bin then you'll have to turn the contents with a fork every now and then to mix and aerate the layers - not having to do this is a huge benefit of the rotating compost bin we have.
In a tiny garden / yard a 'hot bin' system (very speedy composting if you can get the temperature right) or a 'bokashi bin' system (fermenting / pickling kitchen scraps) or a wormery could work for you - these methods also allow you to add a wider range of unwanted foodstuff material from the kitchen.
Composting is a bit like sourdough bread-making, it's a hobby which draws you in slowly and before you know it you are obsessed...