The simplest and cheapest way of getting lower gears is to get a smaller front chainring. But that also gives you a lower top gear. It requires no other changes except maybe shortening the chain.

To keep your current top gear but add lower gears you can switch to 8/9 speeds as you say. Going from 7 to 8/9 speeds you need to replace the cassette, the chain (probably, at least for 9 speed), the derailer, and the gear shifter. You can probably keep the front chainring you have. But there are several important caveats:

You need to know if you have a cassette freehub (good) or a freewheel hub (less good). Your hub is either an SRAM DualDrive or a Sturmey Archer. As far as I know, both are cassette only, but I'm not sure. More info:
https://sheldonbrown.com/free-k7.html

8/9 speed cassettes will normally not fit on 7-speed hubs: https://sheldonbrown.com/k7.html#bodycompat
But you may be able to find an 8/9-speed cassette body that fits your hub, look for it in the 2nd hand market or ebay.

Beware that upgrading from 7 to 8/9 speeds may increase your hub width, which requires spreading the rear dropouts apart, a dangerous operation on a small wheel alu frame. This may be a dealbreaker. Try to measure the hub/dropout width. If it's less than 135 mm, it may not be possible to go beyond 7 speeds safely.
https://sheldonbrown.com/k7.html#up7

If you have the SRAM DualDrive, this has a combination shifter that shifts both the hub and the cassette. You'll need to find an 8/9 speed one, and since they're not made anymore afaik you may have to look for it in the 2nd hand market or ebay. You don't need a special DualDrive derailer, but it must be an SRAM MTB derailer.

If you have a Sturmey Archer hub it's easier, you can choose between SRAM or Shimano style derailer/shifters, just make sure both are of the same type (or if not, verify they are compatible).

If all this turns out to be impossible or too expensive/risky, you may be able to find a 7-speed cassette with more range than the one you have. Count the number of teeth on the smallest and largest cogs, and compare with what's available, e.g. https://www.bike24.com/mtb-cassettes...%5D%5B424%5D=1
NB: If your smallest cog is more than 11 teeth, it may be because your hub doesn't allow that small cogs, in which case you shouldn't buy any cassette with an 11t cog.