I travel to Gatwick often. Your best bet is the northbound Thameslink train from Gatwick Airport Station and exit at Blackfriars Station, then walk to your hotel which will be just across the Blackfriars Bridge (you could take Gatwick Express but you arrive at Victoria Station and then you need to take the Circle line tube to Blackfriars). I recommend buying the Thameslink tickets online in advance because the Gatwick Airport train station is always rammed and you will have to queue to buy your tickets.

Your hotel is well located and you can take the Jubilee line tube from Southwark station to go towards the West End (Westminster, Bond Street, Green Park etc) and for the museums like the V&A you can change for the Piccadilly line at Green Park or the District line at Westminster. London tube signs often work with cardinals (e.g. Jubilee line Southbound), so get comfortable with what is West, North, South, East etc. You should not worry about "express trains" like the NYC subway, these do not exist within Central London.

For the tube, do not fall in the tourist trap of buying paper tickets (more than double price!) or an Oyster card. You can tap at the barrier directly with your credit card or Apple/Google Pay. This is what locals do and it is the cheapest way.