The first choice would be dirt infields and grass outfields hands down. I am a TD that uses mainly dirt infields but have to use a grass infield at one complex. There are complaints every year about that grass infield across all age groups and teams have not returned. The word gets around and even the youngest age groups take the dirt seriously.
An all artificial surface will help get games in but is extremely hot and the ball bounces like a tennis ball. We have had some High School games on the football field and it was better than not playing. Sliding/diving on it left some nasty abrasions. I would be hard pressed to play a midsummer tournament on an artificial surface. Between the wicked bounce and the heat I think we'd find somewhere else.
The option of a grass infield is the worst. The grass effects the play in odd ways especially in the transition from grass to dirt; the most spectacular is shooting a ground ball up into the player's face. Softball will wear big holes in front of 1st and 3rd. Water tends to pond in the base paths and the field becomes unplayable very quickly. and you can not tarp a grass infield without killing the grass should the sun come out.
I work with a local Little League that finally built a dirt infield at one of their fields after 50 years of grass infields. The baseball guys are liking the true bounce, the lack of ponding, no grass lip. Most of their tournament games take place on dirt infields so it amuses me to se the al star teams practicing on the 'softball' field. They are considering converting the remaining fields in the complex to dirt. I have seen some complexes use a portable mound for baseball which allows the field to be truly multi-sport.
Do not under any conditions do a grass infield unless you want to be baseball only. The artificial surfaces are not game ready and would be fine for a minor tournament but will not attract the best teams in the middle of the summer. Good Luck