Welcome to FFXI and Shiva
Here's some generic tips that every new player should know:
One of the first things you need to know is about signet. Talk to the gate guard in your city to get this. Depending on what city, they will have either a I.M., R.K., or W.W. by their name. This allows EVERY mob in the game to drop crystals. Selling crystals early on will fund your character to lv 20-30.
Try to sell as much as you can on the Auction House. You often get more then just selling to NPC's.
Also try to get a Linkshell. A linkshell is a group of people you can look to for advice and help.
Try FableUnity one of the largest and longest lasting linkshells on the server.
www.fableunity.com.
You can also send a /tell to me, Ryhaa, and I'll try to get ahold of a sackholder to give you a pearl.
As for your specific questions.
Red Mage isn't a bad job to start as. Since new characters have limited space you'll have to learn to balance your gear between Black Magic, White Magic, and Melee. Though Rdm's don't melee often in full 6 man parties. The main (non-advance jobs) that Red Mages sub are White Mage (much more usful at lower levels thanks to ~na spells) and Black Mage (provides more MP). If you do start out as a mage, all spells are available at NPC's in your starter town. These NPC's sell the early level spells cheaper then at Auction House.
Haven't played Summoner much but in parties they are normally the main or back up healer. Unlocking the Summoner job ranges from Easy to Annoying (It's mostly determined by your luck getting all the different weather.)
I remember seeing a Diony running around, but I don't know the person and have no idea if they still play.