Might depend how long you are looking at saving for and whether there is a goal in mind or just want to put some away. Funnily enough had the same conversation with my 21 year old last night- he is currently able to set some money aside each month but is looking to go to do his masters in September so will need that money then. If he puts it in savings then he is guaranteed to keep the same amount and with ,low inflation this shouldn't depreciate too much in real terms. He could put it in shares with the chance of much better returns but given the uncertainty over post brexit and the continuing Covid situation could lose a lot.
For example someone investing in UK shares this time last year would be in for a shock in a couple of months and losing a lot due to Covid, I don't think they have fully recovered those losses yet.
By shares you should be thinking funds really so that the risk is managed and spread out. It's interesting picking individual shares but risky, no matter how much research you might do.
If you are looking at saving anything like medium to long term (I'd say at least a year) then usually shares/ funds are going to outperform a savings account. Whatever you do make sure it is in an ISA.
There is a separate FTSE thread where you will find a lot more information, from ridiculous or chancy punts on individual shares to info about some funds that perform well. You aren't restricted to the UK markets if you invest in funds.
www.moneysavingexpert.com