I can't be sure from the photo, but it looks like your hot and cold pipes coming out of the wall might be plastic PEX or copper with soldered fittings. The shut off valves on each look like older multi-turn valves as @HoneyDo suggests. Those valves can be rebuilt in place, which I think may be easier than replacing them for a novice at least. In my experience, it's the hot water side where the internal washer breaks down and pieces can flow downstream and clog a faucet intake (BTDT!).
Assuming the above is the case, you have two issues to deal with:
1. The hot water shutoff valve probably will not fully shut off now, likely missing part or all of its internal rubber washer.
To fix this, shut off water upstream of that valve (perhaps at your hot water heater). Disconnect the hot water supply to the faucet at the connection circled in blue in the photo below (have a bowl and rags ready as some water will be dripping out).
Next, rebuild (replace) the internal washer on that valve or replace the valve. Rebuilding is surprisingly easy. Ask This Old House made a video that shows the process How to Rebuild a Shut Off Valve | Ask This Old House and rebuild kits or just the washer are easily available at hardware and big box stores. Just make sure to remove any remnants of the old washer before installing the new one.
Once you have your valve rebuilt or replaced, make sure it is turned off and turn the water back on where you earlier turned it off upstream. Make sure the valve is not leaking; fix if it is. Once the valve is not leaking, hold a rag and bowl over the disconnected end (blue circle area), slowly turn this shutoff valve on a bit to make sure water comes through and flush anything out, then shut it off again.
2. Now you need to backflush the faucet's hot water side to try to remove any clog.
The idea here is that anything that got pushed into the faucet intake is most easily removed by reversing the water flow and flushing it back out. In this case, we have the intake disconnected at the blue circle. Holding that loose end over a bucket to catch the water, set the faucet to medium water temp (halfway between hot and cold) and turn it on. Some water from the cold water side should flow through to the hot water side and down out the hot water intake, hopefully dislodging any blockage. You may need to put your finger over the faucet outflow to block it and force the water back down the hot water intake tube.
Hopefully you will see or hear a clog blown out and into your bucket. Once you've backflushed, re-connect the hot water supply line at the blue circle area, turn the hot water shutoff back on, and test your faucet.
