Soldering tweezers and soldering pencils are usually way over your budget. Best thing to do is get a cheap hot air station and a soldering iron, so you can use hot air to remove the component, put fresh solder on using the iron, then use hot air to put the new component on. As long as you don't go crazy with the hot air, and cover important BGA ICs, you will manage fine.
I started off with a combo solder iron and hot air station for £40 and was doing fine. Once you get more income doing board repairs, you can reinvest into better equipment.