As for reaching out to recruiters, while I can definitely ask some friends for referrals, do you mind sharing the best way to reach out to companies that I have no connections to?
Follow company pages, look at company employee lists and reach out to HR people with (a more professionally written version of) what you've written here.
I wouldn't use exactly the following, but along the lines of...
"Good morning/afternoon <name>,
I am an <descriptor> studying <field> at <school>, with experience in <stuff> and a strong knowledge of <other stuff>. In preparation of my graduation, I'm reaching out to potential employers to expand my network and, ideally, find a position which suits me at a company where I can use my skills, expertise, and knowledge of <field>. Your company interests me because of <stuff>, and I would love to further discuss what your company is in need of.
Looking forward to hearing back from you,
Regards,
<name>"
Depending on the responses, you may be able to eek out some additional networking contacts by following up with something along the lines of... (in the case that the company is not hiring, or not interested)
"Thank-you for the swift reply <name>,
I appreciate the time you've taken to read and reply to my previous message.
It is regretful that you are not hiring, but I am thankful for your candid response. With your permission, I would like to follow up with you in a few months time on the chance that a position opens up.
With how tight-knit the <field> community is, I was hoping you may be able to introduce me to another <company, person, etc.> which may be looking for an up-and-coming <desired job position>.
Regards,
<name>".
Keep in mind, this is spitballing with no context and may not be appropriate in all circumstances or with all companies.
Follow company pages, look at company employee lists and reach out to HR people with (a more professionally written version of) what you've written here.
I wouldn't use exactly the following, but along the lines of... "Good morning/afternoon <name>, I am an <descriptor> studying <field> at <school>, with experience in <stuff> and a strong knowledge of <other stuff>. In preparation of my graduation, I'm reaching out to potential employers to expand my network and, ideally, find a position which suits me at a company where I can use my skills, expertise, and knowledge of <field>. Your company interests me because of <stuff>, and I would love to further discuss what your company is in need of.
Looking forward to hearing back from you,
Regards, <name>"
Depending on the responses, you may be able to eek out some additional networking contacts by following up with something along the lines of... (in the case that the company is not hiring, or not interested)
"Thank-you for the swift reply <name>,
I appreciate the time you've taken to read and reply to my previous message.
It is regretful that you are not hiring, but I am thankful for your candid response. With your permission, I would like to follow up with you in a few months time on the chance that a position opens up.
With how tight-knit the <field> community is, I was hoping you may be able to introduce me to another <company, person, etc.> which may be looking for an up-and-coming <desired job position>.
Regards, <name>".
Keep in mind, this is spitballing with no context and may not be appropriate in all circumstances or with all companies.