Obviously, grammar and spelling need to be perfect before you send it off, which is why it's a good idea to put the manuscript aside for a few weeks (I usually wait 5 or 6); it's amazing how many errors you can find on re-reading with fresh eyes. You may want to join a writing community like www.mywriterscircle.com (I've been a member for many years and it's helped me a lot).
Few reputable publishers accept unsolicited submissions anymore, but there are a few medium sized traditional publishers that do, with Andersen Press probably being the best. I made a blog post listing them here: http://www.walkerproductions.co.uk/blog/13-reputable-uk-childrens-publishers-accepting-unsolicited-submissions
So going the agent route is probably best to start with. They will sell your MS to the best publisher in return for 15% commission. Every writer should have a copy of the current Writers' and Artists' Yearbook http://www.amazon.co.uk/dp/1408192454/ (the new 2016 edition is out in July), which lists who does what in UK publishing.
Make sure you research any prospective agents thoroughly online before submitting anything. Does their website look professional? Have they successfully sold books in your genre? Most are members of the Association of Authors' Agents, although some reputable agencies are not. Just have a look at what they've sold. Then decide who to send the MS to. The larger agencies have many agents working for them, so you should read their profiles and select the best fit and address your submission to them. Always follow their submission guidelines. Most are either email or form submissions, which is handy (although I have had a few subs and replies lost in the ether from time to time). Usually they ask for the full MS for picture books, either as a Word or PDF file (be aware that formatting may be lost if they have a different version of Word to you). For longer works the first three chapters and a synopsis is usually requested. The synopsis is not a blurb or teaser, but a concise summary of the main plot, including the ending. Try to get it onto one page of A4. Everything, including the synopsis, should be double spaced (select all and press ctrl +2), 12 point Times New Roman, no space between paragraphs unless starting a new section. In other words, the complete opposite of how I've formatted this blog post.
Put your address and contact details, the title, your name and the word count on the title page. Make sure you number the pages. Attach the file(s)to a cover email which introduces you and your work.
Dear [agent's name],
Please find attached my picture book/novel, TITLE IN CAPS (word count). [Short summary of the novel, no more than three sentences, then a little about yourself].
Don't say your daughter and her classmates loved it. Don't say the agent will be missing out on loads of money if they turn it down. Don't say it could be a movie and you will write the script.
Select about a dozen agents and submit to them simultaneously. That is a perfectly standard thing to do, unless they specifically say they only accept exclusive submissions. Do not CC them all in to the same email though!
Expect to wait between 1 and 3 months, and if you haven't heard back send a polite email enquiring whether they received your submission OK.
Everyone gets rejection. Don't let it bother you. They are almost always form rejections that say about how your work is not quite right for their list. But if they take the time to write you a personal rejection treat it like gold dust and treasure it.
If you run out of agents, try the publishers listed on my blog. Then you may want to look at self-publishing (note: use a Print-On-Demand company like Lulu.com where there are no set-up fees. You do not want to be paying vanity publishers thousands of pounds for them to dump 100 books on your doorstep). That's the main thing you need to keep in mind: "money should flow towards the author, not the other way round". You do not want to be paying publishers or agents anything apart from commission and 'expenses' like photocopying and proof copies. Publishers make their money by selling books. Agents make their money on commission from selling books. Do not hand over any money unless they have sold something.