Dee, first you have to follow the rules.
You can't use someone else's list. You can only send your newsletters or sales solicitations to folks who have "opted in" to your list. To be totally safe, you really need to set up the "double opt-in" scenario, where someone says "I want to join", then you send an email to that address and they reply saying "I really want to join, that was me, not some joker who signed me up", them they click a link saying "Yes, I want in" and you are then clear to send.
There are companies who offer these services, as well as mailing the newsletters to the subscribers. Or you can do it yourself. Most hosting accounts offer free scripts, such as phpmailer, that will go through the routine, you just enter the initial link in your signup box.
What you can do to get subscribers is write all sorts of short articles that others may include in their newsletters, or pay to have them soft sell / recommend you to their readers. You can also set up one or more blogs, but it takes some skill to set up the SEO to actually attract people.
Review sites are also a good way to go. Review vacation spots, golf courses, sleep aids, cribs, garden hoses, whatever fits the demographics you are chasing. 4 or 5 "reviews" and several signup boxes, even (dare I say the words?) pop-ups or pop-unders asking them to become subscribers. You can even throw in some Google ads and maybe pick up a few commissions.
Which brings us to the bottom line. What is your motivation for this? To "get your name out?" You"re a kind person and want to share? Or, you are looking for some income as a result of your efforts? Probably the last. So, how are you going to (I hate this word) "monetize" your newsletter / website / whatever?
I could go on and on, but I'll stop there. Any more and I've have to start monetizing myself.