Definitely get a water bottle and cage -these just attach onto the frame, all you need is an allen key to put in the bolts. It's really easy to just reach down to the frame and grab a drink whilst you are riding and you must be carrying some water on ten mile rides I'd have thought.
Also 100% get a track pump -
something that looks like this - which has a pressure gauge on it, and make sure you keep your tyres pumped up to the proper pressure range. It'll give you a range on the side of the tyre and you should keep it within this range. what the PSI is depends on the tyre so you need to look. As a rule of thumb, if you squeeze the tyre then it should be hard like an apple, not soft like an orange. If your tyres are under pressure and you pump them up, you probably won't believe how much easier it is to cycle. It makes a massive, massive difference.
The Joe Blow one I've linked to is a good pump but you might find cheaper ones that are just as good, I don't really have the experience to recommend a specific one.
With the chain, you should be cleaning and oiling it yourself, weekly or monthly depending on how much you use the bike. This will help to keep the bike working smoothly between services and reduce the amount of wear and tear so you don't have to replace parts so often. I'll be honest here and say even as a daily commuter I rarely did this, let alone monthly but it's something you definitely should do and I know my failure to do so has cost me money through needing to replace chains/cogs that wouldn't have worn so quickly if I'd actually maintained my bike.
You need a rag of some kind to clean the chain, a can of GT85 to remove any rust and protect against water ingress and some "dry" chain lube/oil to put on the chain - if you start riding in the winter/rain then you will want "wet" chain lube/oil to use instead.
There will definitely be youtube videos that can show you how to do this, along with how to change a punctured inner tube but I would also suggest doing a search for some basic cycle maintenance courses, as many councils offer these and you might prefer to have someone teach you rather than learning from a youtube video.
For the issues with your gears not switching smoothly or going into the right places you want a service really - this is most likely just an adjustment that needs doing (called "indexing gears") which you can do yourself with a screwdriver if you want to try to learn from youtube videos. Personally I have been taught how to do this and suck at it and decided long ago to just hand my bike over to a pro to get it right.
You are fine to just use the back gears if that's all you need but a service will get the front gears working and help you get up hills