Can you climb without climbing shoes? Sure - in many places, your regular sneakers or hiking shoes/boots will work. This is generally limited to climbing walls with screw in climbing holds or real rock walls up to maybe a 5.6 rating. But for more technical climbing on both walls with artificial climbing holds and rock walls with natural features, climbing shoes make a huge difference and you may not be able to climb without them.
Which shoes do you get? Well, pick ones that fit. The only way you can know if a shoe fits is by trying them on. Don't trust shoe sizes, because they are all over the place. Go to REI or another climbing gear store and try on different shoes. They should fit snuggly without socks on.
When choosing a climbing shoe, understand that they are are made with different bends.
These are good up to 5.10 and V2 routes
Flat, comfortable and affordable
If you really get into climbing and start climbing 5.10 and V3 - you will need different shoes
La Sportiva Tarantulace
Black Diamond Momentum
Scarpa Reflex V
Evolv Defy & Elektra
Butora Endeavor
Five Ten Niad Moccasym
Shoes are flat
Midsole may be medium to stiff
These are great for all day use
These have a light downturned shape (camber)
Helps feet gain more powerful and stronger foothold
These aren't as comfortable as neutral shoes
Better for cracks than aggressive shoes
This is more shoe than you need as a beginner and will be uncomfortable
These have an aggressive downturn
Preferred for overhang climbs
Really uncomfortable - not for all day use
Shape doesn't fit into cracks as well as neutral and moderate shoes
NOTE: for beginner climbers, these won't make you a better climber, they'll just cause pain
These aren't the shoes you are looking for