HELP! Complete Novice - Chainring / Cassette advice

As others have said, you can get a reasonable idea whether your chainrings need replacement by comparing their teeth with new rings. Pointy or 'shark fin' teeth suggest replacement is required.

As others have also said, it may also be an adjustment issue (much cheaper to resolve than 'ring replacement!). One way to check is to compare how the gears function on a workstand versus under the much-heavier load of your leg-power. If they're running sweetly - smooth and quiet - on the stand, changing crisply up and down the rings and sprockets - but still skip when you're applying power when riding then it's a problem with the parts (possibly worn chainrings but perhaps also incompatibility between the chain/cassette/chainrings - do you know for sure that they're supposed to work together?).

If you're new to this stuff then you probably don't have a workstand... You can get a handy little one which lifts the rear triangle tubes on hooks quite cheaply (and you might currently find a more substantial item at Lidl!) but an improvised alternative is to put a broom handle through the top of the rear frame triangle then rest the broom across the back of two dining chairs (or similar). This keeps the rear wheel off the ground so you can turn the pedals with one hand and operate the gears with the other. Works remarkably well!

your rear derailleur will need the setting up, pretty easy, google - park tool rear derailleur YouTube - it's 16 minutes long, all you need is a x head screwdriver and your fingers (skip the pulling the cable tight and undoing the anchor bolt, but if the setting it up doesn't fix it you might have to start again with doing the cable setup)

The thing with the bicycle is that changing parts is easy; you look at a couple of YouTube videos, pick up a hammer and off you go.

The problem lies with deciding which parts to change.  For such a simple machine it can develop a thousand squeaks, rattles, skips and grumbles and unless you've some experience you can waste time and money chasing your tail.

Find someone who knows a bit about the velocipede, we are everywhere, ask for advice, we all love to show off our expertise, and only then rush out to buy the bits.

Best of luck.

Take it to your local bike shop.

I purchased a new chainring as I thought I had bent mine, given it had a developed a wobble. Took everything apart, transpired it was the bottom bracket, chain ring was fine. Had to borrow tools to change the bottom bracket. Would have been easier, quicker and probably cheaper to get someone with a good knowledge to check it and fix it.

Are the chainrings looking sharp, with uneven wear, like sharksfins.

Compare them to pictures.

Worn chainrings tend to cause chain slip when laying down proper power, pulling away from lights or sharp inclines.

My middle ring (your smaller) wears fastest.  I even had to obtain a new granny ring recently, though the worn one caused a weird chain suck thing rather that the slipping that the middle or large ring causes.   It would drag the lower chain up towards the frame and cause a "lock"  Really bad when on a new chain.

New gear cables and setting up right would be a cheaper thing to do first, and seeing the age of the bike, probably needed too.  At the very least a good lubrication.

Could be any number of things really.

I'd double check that the chain is compatible with the cassette (e.g. 9 speed, 10 speed?), then check that the rear derailleur is shifting nicely between all the gears - if not, replace the rear derailleur inner cable which involves re-adjusting the derailleur. Also check the jockey wheels still have some life left in them.

The best bet is to take it to a bike shop, but there's plenty of youtube instructions if you want to spend some time to do it yourself and you shouldn't need too many tools (a good cable cutter is recommended for replacing the gear inner - ordinary pliers will leave frayed ends).

Chainrings do wear, but they usually outlast several cassettes.  It is more likely to be an incompatible chain.  Check that its a compatible manufacturer and the correct gauge (speed) chain.  

Chains and cassettes are pretty complex these days and are engineered to the correct cassette / chain ring width with specific angles to match.  Get these wrong and it will slip because its not seating properly and therefore not engaging.  

Chains come in 9, 10, 11 and 12 speed, the most popular being 10 and 11 speed. The 10 and 11 are not compatible, the latter having a narrower, more profiled guage.   

The alternative would be to buy a modern groupset.  The best value is Shimano 105, 11 speed.  

You could fit new chainrings but still have the issue, because the issue may be being caused by the derailleurs, cables and/or indexing.

My advice would be to take it to a reputable bike shop who should advise exactly what needs doing to get it running sweet again. Without seeing the bike you can't confidently say what you need to replace/do.