This is a bit loaded question, but I will do my best to address your question.
For a 75 gallon tank, you can run a hang-on back skimmer and you should be fine as long as you purchase a high quality one. Currently, the best hang-on back skimmers on the market would be Reef Octopus and AquaMaxx. You can have a pretty successful tank without a sump. A sump will provide the advantage of being able to hide things, more volume, and more flexibility of the equipment you want, but I would agree adding a sump is intimating because of plumbing requirement. These days, you can purchase an all-in-one aquarium with compartments to hold skimmers, heaters, auto-top off systems, etc in the back of the tank, but they will be more expensive than if you built your system with a sump.
For a sum up -
For cheapest setup and size -
First -
For cheapest aquarium, there are going to be two options - purchase a 75 gallon fish tank from Petco (brand name is Top Fin). This tank is around $199. Or you can purchase the combo with the stand for $299.
Second -
Find a used tank from Craigslist. You can get a pretty good deal from there - especially in the summer when most aquarium keepers leave the hobby. If you buy a tank from Craigslist, you will need to clean the tank with distilled white vinegar to get it ready for usage.
So all that being said, the least expensive setup would be:
- Purchasing a used tank setup from Craigslist
- Purchasing whatever missing equipment you need with lighting being your most expensive
- Going fish only or fish only with live rock
- Purchasing frag corals from local reef tank owners instead of fish stores
Buying a brand new -
If you chose to purchase brand new, there are several pieces of equipment you will need just to get started. Here is the main list. Your options and price will increase depending on your long-term goals:
- Protein Skimmer
- RODI System (A must if you choose to keep a reef tank)
- Heater
- Salt Mix
- Test Kits
- Lighting (Will get more expensive if you chose to have a reef tank)
- Biological Filtration (Either with Canister/Wet/Dry if you choose fish only or porous dry rock if you chose reef tank)
- Flow (Powerhead for circulation and oxygen exchange - again gets more expensive if you want a reef tank)
- Substrate (Aragonite based)
- Refractometer (better than hydrometer)
For a 75 gallon tank - you are likely looking at ~$1000+ all in even for a cheap setup. As for fish, 5-7 fish should be fine in a 75 gallon tank, but you have to make sure you put the best compatible fish and likely easy to keep as you are beginner.
Check out this list of top 10 saltwater aquarium fish for any reef tank blog post for a good list of fish to get you started. You can also check out this guide for saltwater aquarium selection to get additional information regarding aquarium types and the glass vs. acrylic debate.
