I hope you don't live on the top floor?
Your landlord should (is required to) have had your gas fires tested and certificated, you might like to see the certificates to verify they are up to date and appear to have been done by someone reputable. This is relevant so you don't wake up dead one morning with carbon monoxide poisoning.
Try to work out what rate you are being charged for gas and electricity if you don't pay the supply companies directly - some landlords like to chanrge tenants a higher rate.
This is relevant because gas is usually cheaper to run than electricity, subject to the price you pay being normal.
The cheapest and easiest thing you can do is draughtproofing. You can get various strips from your local DIY store. If you are buying a lot it is cheaper from Screwfix dot co dot uk, but bear in mind the postage cost.
These strips will not stick to dirt so clean the paintwork of the window and door frames before applying it.
Bubble wrap is rather unsightly on windows, you can buy clear plastic sheet that looks much better, and is invisible if you have net curtains. It is attached with double-sided tape and you will do a better job with 2 people hanging it, as if it's wrinkles or uneven it is difficult to peel off.
Also - do your best to keep the flat dry. No wet washing in the rooms, use an extractor fan (if you've got one) in bathroom and kitchen. It makes a big difference.
Unless you've got a bad chest, you won't need heating all night in the bedroom - buy a couple of electric plug-in timers and non-fan elctric heaters, and set them to come on before going to bed/getting up and to be off while you're out. Put them well clear of furniture, curtains, clothes etc. You can take them with you when you move. Don't use plug-in adaptors in clusters on the sockets, use a 4-way adaptor with a fused plug and a short lead. very cheap in Tesco, like the timers. If the bedroom's cold, think about an electric blanket - not very cheap to buy, but economical to run, and again, not a fixture, and it will keep the bed warm and dry even if the flat isn't. Avoid buying second-hand or market stall electrical goods.
(edited to remove rude comments about your home - sorry)