It's the purchaser who pays stamp duty. Full guide here:
https://www.moneyadviceservice.org.uk/en/articles/everything-you-need-to-know-about-stamp-duty
Other costs of buying here:
https://www.moneyadviceservice.org.uk/en/articles/estimate-your-overall-buying-and-moving-costs
When selling, the first cost you'll probably encounter is paying for an Energy Performance Certificate. (It's a legal requirement to have one before you place a property on the market):
https://www.gov.uk/buy-sell-your-home/energy-performance-certificates
Then you'll probably have estate agent's costs to pay. (I write 'probably' because you don't actually need to use an estate agent. You can do the job yourself but that won't give you access to websites like RightMove.co.uk, where most house hunters will be looking).
Traditional high street estate agents charge commission on sales, meaning that you only pay when you sell (but you could end up paying quite a lot). Web-based services such as Purple Bricks charge a flat fee to put your property on the market (which is usually considerably lower than you'd pay in commission) but you have to pay them whether they sell your house or not.
The only other costs involved in selling are legal fees. While it's possible to do the conveyancing yourself, most people engage the services of a legal professional to do the job. Licensed conveyancers can be a lot cheaper than solicitors but it seems hard to find one these days (because solicitors seem to buy them out to reduce competition).