How do you rank for major keywords...
So here's the deal.
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I currently rank #1 for "city + keyword" name for a few places

Each have their own landing page which includes the city name and keyword in the title.


HOWEVER

I'm looking to start ranking just for the main keyword

For example, let's say I am currently ranking for texas maid service

How, will I begin to rank for "maid service"?
Write reviews with it your google reviews, do citations and repeat the process that should help you rank it
The same way you rank now just more work .

use ahref and beat all computition stats that it
Create a page on main keyword:

Content more then competitors.
Perfect internal linking
Give social signals
Buy high quality guest posts on homepage and that page.
Boom you will be ranked soon.
citations ann google reviews
Yes, can rank!
You only need more efforts with solid links!
Let's take your example only - 

texas maid service - Medium Competition - You already ranked for this
maid service - High competition because - You need to rank this - Add content to your website focusing this keyword. Do all the backlink using this anchor text. It will take time but if you consistent, you will see ranking improvements.
Awful “advice” given so far. More competition means more links, you already know the rest.
Let's take your example only -

maid service - High competition because - You need to rank this - Add content to your website focusing this keyword. Do all the backlink using this anchor text. It will take time but if you consistent, you will see ranking improvements.
Should it be linking to the home page, or should i make a new landing page dedicated for "maid service"
You probably need to make a page dedicated to 'maid service' and be careful and not confuse google with the other keywords you're ranking for. Just be aware that google now is more aggressive with pushing local and country related pages. This will be so tue with a keyword like 'maid service' if someone is searing for 'maid service' located in New York, google will automatically assume that they are looking for something local and push local pages first. A few days ago I was checking a general keyword connected from a Canadian VPN and the first 3 or 4 pages had .ca extension which was basically useless for the purpose of my research.

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