I have a TRS232 RS-232 Driver(max232 equivalent). On the data sheet, it shows a capacitor on Vs+ and Vs- (1uF)and they just have arrows saying they go somewhere. Can I omit these connections and capacitors and if I do what are the side effects or issues that may arise from this omission?
This transceiver conveniently includes a charge pump to create the proper voltages for RS232 signaling. All four capacitors listed (as well as the standard decoupling cap) must be connected for the device to operate.
If you omit these capacitors, the charge pump won't work. Your positive logic level referrence will oscillate at about 400kH between drawing power from 5V through a diode, and getting a boost to about 10V, depleting the charge in C1 or C2 (I don't know which). The inverting supply will be even worse - I'd guess it will jump between 0V minus a diode drop and -5V until the capacitor is depleted, but I'm not really sure without knowing the topology of the charge pump they're using. When your logic level reference is jumping around like this, the chip will not be able to determine what is a high voltage and what is a low voltage, and it certainly won't be able to transmit anything. Worst case is that a signal on the RX pin will try to power the chip through the ESD protection diodes because the charge pump isn't working, and fuse the diodes into a conducting state, destroying the transceiver.
The datasheet says that 0.1uF capacitors may be used, but lists 1uF capacitors as the recommended value. If you must attempt to transmit or receive immediately on applying power to the chip, you'll want to go with the 0.1uF caps, but if you can wait for a millisecond before doing this, 1uF caps will give you a more stable voltage. Larger values than that won't help much.
The 'arrows saying they go somewhere' are just voltage markers. You don't need to connect these to anything. If you wanted to test the operation of the charge pump, you would probe these points. You might also be able to sneak off with a little bit of current from these pins if you had to for some other purpose, like powering devices like opamps that require differential supplies, though you'd need to do some extensive testing to be sure that you didn't draw too much current. A buffer powered from 5V would be a good thing to have.
Definitely required. Your drivers won't output anything without them. In this answer I explain how the charge pumps work using these capacitors and also why it's not a good idea to use 10\$\mu\$F instead of 1\$\mu\$F.
Those capacitors are essential. If they are omitted the charge pumps won't work properly.
The +8.5V and -8.5V outputs can be useful for powering devices that need two supplies, such as op amps. I've used them for that purpose, enabling the whole system to be powered from 5V.
You also need the decoupling capacitor on the Vcc pin.