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I have uploaded my first website and trying to learn how to get search engines to see it. Google has a manual verification string to insert into the root dir but I'm not sure if it is for the dir or the home page on the root dir. If it is the home page, using html meta tag and placing the character string from google should this be all I need to do. this is to verify my site to google

My other question is : are different host servers like godaddy or bluehost crawled more than my server which is a small independent hosting company and are there any tricks or methods to more quickly get my website found

You can insert custom meta tags in the 'Meta Tags' section of the page properties.

I do not think host servers like godaddy or bluehost crawled are more than a small independent hosting company.
The SEO results are mostly affected buy the content of your website.

Google has a manual verification string to insert into the root dir but I'm not sure if it is for the dir or the home page on the root dir.

The verification code file should be placed in your root directory along with your other files.

are different host servers like godaddy or bluehost crawled more than my server which is a small independent hosting company

No, don't believe so. Even if they did they would never admit it.
However, your host can have somewhat an impact. For example: site speed (does it load fast), downtime (is your site down a lot which may impact the crawling of it) and backups (make sure your data is backed up in case it hacked),

are there any tricks or methods to more quickly get my website found

I suggest the following (note: not an exhaustive list):

1. Finish the process of verifying your account.
2. Create a webmaster site with google and Bing and upload your sitemap to them
3. Create your business listing with Google and Bing
4. Make sure you have good content on your pages
5. Make sure you have an ssl certificate for your site.
6. Make sure it is mobile friendly
7. Promote yourself on free sites (Facebook, Yelp, etc.)
8. Entice customers to leave you reviews on Google, Yelp, Facebook, TripAdvisor, etc.

A lot of my clients ask this question and there is not necessarily a simple answer but I tell them simply having a website live is not going to drive thousands of customers to your business. It will help and will start showing up eventually but it never hurts to put in a little time promoting yourself and using any free avenues you can to help (ex: Facebook). Good Luck

Thanks this is a new area for me

Also try to gain backlinks which is still a mystery for most of us and does have an impact SEO search result rankings if you are looking to be a professional some day.
For more information and simplicity check this blog out: https://backlinko.com/hub/seo/backlinks