It totally depends on what it is you're trying to target with your serum. If you're trying to target fine lines or ageing, I always think it's best to go straight to the only proven thing to help - retinoids (like tretinoin) or a more gentle derivative like retinol which you can ease into. If you're looking to even out your skintone, brighten it or fight pigmentation, vitamin c is best. However, if you're looking for something hydrating, hyaluronic acid is always the way to go. The Ordinary have one for about a fiver, but I personally think The Inkey List's is nicer and for around the same price. For ultimate hydration, there is Hada Labo Gokujyun Premium Lotion - a Japanese hyaluronic acid with zero fragrance or alcohol which would be good for sensitive skin. You can find it easily on eBay, however right now the new US/UK version of this product has dropped and is on offer on Superdrug for just under £12. 'Hada Labo Tokyo Super Hydrator Lotion'. This may be of interest to you as you have dry skin. The Ordinary's 'Buffet' is a great all-rounder serum and at a really affordable price given its benefits. The Ordinary's Niacinamide 10% + Zinc 1% is also a great serum if you struggle with large pores, uneven skintone or blemishes.
I always find it's easier to get serums for each 'problem'. So I'll use vitamin c in the morning along with hyaluronic acid, and then in the evening I'll use niacinamide if my skin is acting up or just something more hydrating if I'm using tretinoin.
Some serums off the top of my head that are great:
Manyo Factory Bifidalacto Complex - This stuff is raved about on reddit. People say it's a dupe for Estee Lauder Night Repair. Packed with lots of fermented ingredients, niacinamide and copper tripeptide. Great for plumping the skin.
NIOD Copper Amino Isolate Serum - This just gives an overall more youthful complexion. It's great at repairing the skin. I can't even pinpoint what's so special about this stuff; I just love it and think of it as liquid gold. A wonderful product as is NIOD's 'RE: Pigment'.
Dr Jart Cicapair Tiger Grass Re.Pair Serum - Amazingly calming. If you deal with sensitivity or any kind of redness, this is heavenly.
COSRX Snail 96 Mucin Power Essence - The first ever Korean skincare product I bought. It gives beautiful hydration and makes your skin feel so plump. Can be good for evening out skintone, too.
Superdrug Simply Pure Hydrating Serum - I'm sure everyone knows this by now. £2.99 and the one thing I will buy in bulk because it's seriously just that amazing for the price.
Garden of Wisdom's serums (found on Victoria Health) - Alpha Arbutin 2% and Kojic Acid 1%, Super Bakuchiol.
Paula's Choice booster serums.