Some additional tips to add to what's already been suggested:
youTube is worth checking out, as there are some very nice, easily digestible short tutorials to get started. Believe me, once you get the Process Mapping bug, you'll be addicted (My name is Daisychain and I'm a business process addict 
Heres a collection of process mapping tutorials (I hope this link works)
Visio (or equivalent free-ware) make short work of drawing the boxes and arrows, or building swimlanes. If I were in your situation I would be putting in a business case to your manager to get a licence. If your company uses O365, you might be able to get it added onto your corporate license so others get to use it and build the skills wider than just you (gives you a stronger case to invest).
Simplistically, in each of your process boxes, you should restrict the description to a verb (active, doing word) and a noun (a thing) eg:
- Review business case > update financials > submit business case > approve business case
- Enter order details
- Ship order
- Issue Agenda
- Capture actions
- Allocate responsibilities
if your process gets to a point where a decision, a choice eg different actions or options need to be taken, use a diamond shape and have arrows coming off the diamond points eg Yes (if Y, go in this direction) or No (if No, go in this other direction). Get a colleague (ideally the subject matter expert) to read out your process and make sure they agree with the logical flow. If not, get them to tell you what is missing or wrongly stated so you can correct it.
You will never get your process right first time, that's OK it will always need to be tweaked. That's the point. Make a start and build on it. Set up a workshop if you need several people involved and inputting into your process.
Always give your process map a title to increase understanding and context "Sales Order Process" "Financial approvals Process" "Paperclip purchasing Process".