First off, welcome Fritsy to the wonderful world of Cosplay!
There have been several detailed thoughtful responses, you guys said pretty much everything I was going to say, but said it better than I would have.
I might offer a slightly different perspective as I didn't start Cosplaying till I was 58, I'm 63 now.
For the first couple years I only did one character, Indiana Jones' dad (as you can see in my avatar). There were several advantages to this:
1) The character is widely recognised and popular to the point of being beloved by many. Thus the response from everyone both inside a Con and in the outside world was positive. I have never been made fun of, harassed, or had any negative reaction.
2) The character has an iconic look, so my first efforts (pretty much Closet Cosplay) though simple and inexpensive were recognised by all. As I continued with the character I upgraded the costume piece by piece eventually arriving at a very nearly Screen Accurate costume. So I didn't have a large initial financial outlay.
3) The costume is pretty much ordinary clothes, meaning that I could ride Public Transportation, wander in and out of Conventions and more or less blend in with the public.
4) The costume is extremely comfortable to wear, has comfortable shoes, and involves no mask, helmet, armour, wig, prosthetics, or cumbersome props. Being in my 60s now comfort is of prime importance. At Conventions I'll be in costume for 8 hours or more and it's no problem.
5) Dr Jones Sr doesn't carry any sort of weapon. This is a huge advantage going in and out of Cons and wandering about in public.
Other points I might mention:
1) It's nice to have a useful prop at Conventions. My Dr Jones Senior outfit includes an umbrella, which has shielded me from rain in Seattle and blazing hot sun in Arizona. Besides it gives me something to do with my hands when posing for photos. Another useful prop is a cup, so I can sip water as I walk around a Con and stay hydrated, or have coffee in the morning, or beer later in the day!
2) I feel it's critical for my costume to be very sturdy, with no bits which can get knocked off or broken. As people have mentioned your costume will take a beating! Even then I have duplicates of some items in my boot bin (see below)
3) Due to me being forgetful, I put EVERYTHING connected to a particular costume in a big bin, and when I'm off to a Con the bin goes in the boot. Therefore it's impossible to forget anything.
4) About weapons, yes to what Jediguy said. You have to check every Con, they're all different. One thing I've done is make props that have no metal whatsoever and no moving parts. I now have a lightsabre, a revolver, a dagger, and a sword all of which can pass through a metal detector. The dagger and sword are one-piece foam props, the scabbard and handle one piece, with no blade at all.
5 years into cosplaying, 6 characters now, all of them comfortable to wear all day, the only makeup is colouring my beard to various extents.