"Upgrading" the chain won't be noticable unless you are a very serious cyclist. However, you should make sure to replace the chain when it gets stretched, because riding a stretched chain will cause your other components (cassette, chainrings) to wear out faster and give you more expensive repairs later. If you replace your chain periodically as needed, you'll ultimately safe money and problems.
A chain should last maybe 1,500-2,000 kilometers or so, but it could be more or less based on a huge amount of different factors.
A chain should be replaced once it reaches 0.75% stretch. You can either get your own chain measuring tool, or stop by a bike shop every so often and ask them to measure your chain for stretch (they will do this for free; if they try to charge you just for measuing it, tell them to fuck off and go to a different shop; and don't let them sell you a new chain if yours is under 0.75% stretch, but definitely don't let it reach 1% stretch cause then you start fucking your cassette)
As for "upgrades" to the bike. I would be more concerned with getting a helmet, lights, floor pump, lock, water bottles and bottle cages, and a small repair kit (spare tubes, portable pump, tire levers, multitool such as Crank Bros M17)
A good floor pump is important cause tires lose air just from sitting so you should be checking your tire pressure before every ride or at minimum once a week. Infllate to the pressure recommended on the side of the tire.
These accessories/tools are more important than part upgrades on the bike.