Upgrades not up grades

If you are going to replace your wheels then defo replace your brakes for at least 105. Cost you about £60 for the pair and will save your life.

To take a different spin, the best 'upgrade' I ever bought was a bike computer and a heart rate monitor.  It introduced me to the world of structured 'zone' training and riding which has made a massive difference to my cycling.

But to add to the above, second hand is a great way to go... my brother got a lovely carbon Ribble with Campag Athena 11s for not alot more than £300, but clearly you need to be diligent on the seller.

If you're patient you can find a Spesh Allez/Giant Defy/Trek something or other on Ebay, maybe 2 years old for your £300. That's where I'd put my 300 quid

thelawnet hit the nail on the head.

For what my opinion is worth like a few others I would save your money for a different bike. Much cheaper and better value for money, even in the medium-term.

Also do you have the correct tools or will you be paying for a mechanic? If the later that's another saving with another bike

You wouldn't upgrade that bike. Tiagra is not that much better. Big waste of money

Things to do with a bike like that:

Different tyres as per your riding requirements

Lower gears if you struggle up the hills

Maintenance (new cables, chain, brake pads, etc as needed)

OP, what is your budget? Apart from not having a grand to spend on a new bike, how much have you got? 

Changing from Claris to Tiagra will cost what, £300?

Most experts say the best bang for buck upgrade is wheels, if you really want to change something look at getting some wheels in a sale or on special somewhere.

Like Simon E says, I would wait until you have more money, and get a whole bike in a sale, or even off Ebay.  There are loads of bikes for sale bought by people who want to take advantage of cycle to work schemes, or have seen all the other folk whizzing around on a Sunday and want to give it a go; the bikes get ridden 2 or 3 times then left in a shed or garage, in nearly new condition.

The latest iteration of Claris is rather better than its predecessors. The difference to weight etc of upgrading will be undetectable while the dent in your wallet will be significant. All modern wheels are compatible with 8-11 speed cassettes though some older designs may not work with 11 speed (the hub width for 8, 9 & 10 speed is the same).

Unless you've worn out your drivetrain I would put the money to one side and buy a discounted new bike later in the season. You'll get a better and lighter frame and a decent groupset without spending a fortune.

In the meantime buy some nice tyres.

I would also recommend tubeless but my experience is that Hutchinson Fusion 5 11 Storm tyres are much better than Schwalbe tyres, they roll as nicely but are less prone to damage.

But.  And it is a big But.  By the time you buy wheels, tyres, sealant, valves and possibly rim tape converting to tubeless is expensive.  Add to that the cost of a new group set and you are suddenly looking at an expensive upgrade.  Have you thought about something like the Ribble R872 Disc bike on interest free credit?

Have a look at their website and bike builder (i cant paste a link here, this site wont allow it) or other on line retailers may also be doing interest free options.

My usual advice is to buy with your heart, buy the bike you like (assuming of course that it fits you) and that you will want to ride, a "bargin" which sits in the shed is not a good buy, something that you want to get out on and ride is a wonderful purchase.

Your bike spec

Weight 11.2kg
Gear Shifters Shimano Claris / 2x8 speed
Rear Mech Shimano Claris / 8 speed
Front Mech Shimano Claris / 2 speed
Chainset Prowheel / 50/34T
Cassette / Freewheel Shimano HG50 / 11-28T / 8 speed

If you change the wheels the weight will drop to something like 10.5kg.  

Struggling on hills ?

Changing the groupset and cassette to 48/32 or 46/30 and 11-32 cassette will make a huge difference in the rides for hills. 

The bike has a cartridge bottom bracket(tapered cranks), so this will need swapping at the same time as the chainset-not an issue, just be aware

A decent set of tyres Schwalbe/Continental will make the bike feel nice, especially if you convert to tubeless-wait for the naysayers.

I'd upgrade wheels/tyres before anything else, it'll make the most difference by far.

Something like Campagnolo Zondas or Hunt wheels will lighten your bike and make it feel more 'zippy'. Match with a pair of conti GP4000's or 5000's too!

Tiagra is great; I used it for all-weather commuting for years and it's as near faultless as makes no difference.  It may be a little heavier than 105 but you won't notice that until you pin a number on your back.

If the wheels are more important to you Shimano hubs are super.  Long lasting and easily serviced they usually give sterling service.  Just check before you pay your money that the hubs are suitable for your groupset.  As you say, some will take several different cassette standards, often using spacers.  If there's a wheel builder local to you it may pay to ask for a quote, some will beat the big store prices.  And their advice comes free, which is a pleasant change.

Best of luck.