Index funds do indeed work on Hargreaves Landsdown. You can see some of the Vanguard offerings through HL here: https://www.hl.co.uk/funds/fund-discounts,-prices--and--factsheets/search-results?investment=vanguard&x=0&y=0 (I only use Vanguard because you mentioned it, there are lots of other options worth considering).

Without knowing your goals and situation it's hard to say what you need, but if you're generally inclined to save for some future goal on a long time scale (5+ years), an index fund would probably be a good place to start. Ultimately, you should pick whichever funds you think will do best. If you really have no idea, there are Global All Cap funds, which allow you to broadly invest in companies all around the planet, so you effectively make a bet that globally there will be growth. I think most would consider it likely that, long term there will be global growth. But better returns could be achieved if you pick indexes which outperform the global market.

When picking a fund, you also have the choice of accumulator (Acc) and income (Inc) funds. The difference between these is only that Acc funds automatically reinvest any dividends you might receive, while Inc funds will pay them out to you so you can reinvest (or withdraw) them yourself. You'd probably be better with an Acc fund at this stage, but I think efficiency can depend on local tax law.

As for fees, withdrawl charges, etc, this will vary fund to fund. You will also pay a fee to your broker. It's worth reading the information sheets HL makes available for each fund, and HLs website, to find this information.

Finally, when you've picked a fund, you put your money in and let it sit. You might forward your money to your broker, HL in this case, and then have to tell them where to invest it. If you give money to your broker but don't invest it, it won't get the returns you're hoping for.