I like the way you think.
You too.
A great push-up variation is to place the hands in a position slightly ahead of the forehead. Killer ab exercise, I can’t do many from that position unfortunately.
I trained for years without a bench and managed to grow allright. Pushups are almost without fail categorized as a lightweight excercise that one grows out of quickly. Noted strength training expert Mark Rippetoe notes that pushups would probably be better for chest and arm development than the bench press (I agree) if only they could be loaded to similar levels. However, pushups can be loaded with substantial extra weight and made more challenging in other ways, too. I make mine with my feet and hands elevated (feet on a stool, hands on cheap pushup handles available anywhere) for true full range of motion, and don a backpack with weight plates in it for an adequate load. It’s been a long time since I measured the resistance with this setup, but bodyweight lbs. is easily attained. It’ll take a good while before doing sets of ten full-range, actual-BW pushups become too easy. Also, even now that I bench press, I feel weighted pushups much better in my pecs and triceps and they don’t aggravate my shoulder injury like BP easily does, no matter how deep or heavy I go.
I did use a gym ball as a bench for a while, but stopped after rolling off the ball with two 80# dumbells aloft. I was lucky to escape without a serious face/head injury (even regular bench pressing kills a couple of lifters every year). Another pushup advantage: no risk to life or limb.
Old-timers used to bench press, flye, skull-crush etc. on the floor all the time. T-nation writer Christian Thibedeau even has an article on floor benching and similar excercises. He thinks properly done floor lifts - although lacking in ROM - compensate by their momentum and stretch reflex-sucking qualities: every rep starts from a dead stop, leading to maximal muscle fiber recruition. Certainly a fine plateu-busting technique, if not more.
You should also give some consideration to buying a set of kettlebells instead of dumbells. I’ve been working out at home with a pair of 16 kg kettlebells for just under a year. I supplement them with a light set of adjustable dumbells. At first the kettlebell exercises are a little odd if you are used to the more common dumbell/barbell movements. I will testify, though, that they give an intense workout and build a type of dynamic whole body strength that isolation exercises do not. My old wrestling coach would most definitely have approved.
Scumpup are you familiar with Brooks Kubik? He wrote a book called Dinosaur Training which covers a lot of the old strongman routines. He’s kind of an ass but his training methods are solid. I can’t link from here but I believe his web site is called Dinosaur Training as well. Your wrestling coach would assuredly approve.
While reading this thread I got curious to know what the current WR was for the Bench Press and found the wiki for the history of the record. Turns out the record in 1898 was 361 lbs. Now I now where I’m going if I ever get a ride in a time machine; late 1800s to take a run at the record 
Those Craig’s List ads mentioned earlier hint at the truth of the matter. Most benchs go unused.
If the OP can do pushups keeping with a satisfactory schedule (say, 4x a week) and really keep to it, then it may be time to look into more equipment. A very easy trap to fall into is ‘Pushups are hard. If I had a bench, working out would be easier and I’d do it more often’. The downside to pushups is that eventually, they get too easy. The bench is only there to make the pushup-action more difficult. If you are still at in in a couple months simply using stuff around the house, then consider further expenditure. If you’re not able to do 20-30 REAL pushups, then that bench is just a fancy laundry display rack.
The usual reaction to starting a new workout routine is to drop a ton of money (home equipment or commercial gym) at the frontend without any indication that there will be followthrough. You shouldn’t buy a $1800 guitar before your first lesson nor should you go shopping for steelmills when thinking about starting a business. It’s not sexy but this is something to be eased into. Unless you enjoy taking massive losses on Craigslist.
You’ll need to get one, or a gym membership eventually, but you can do a lot without weights.
- 1-handed pushup
- pushup-handstands (against the wall)
- chair (sitting against a wall)
- 1-legged squat
- 1-legged calf raises
- Pushup between 2 chairs (so you can go lower than regular pushup)
- decline triangle pushups (put your feet on a chair and hands next to eachother to make a triangle)
- dips off a chair (put your feet on other chair)
- putting someone on top of your back for pushups is fun
- loads and loads of ab/core work I could write pages of that
- if you don’t have any weights grab something heavy like paint cans and jump as high as you can repeatedly
- if you aren’t above looting a dump, get gigantic tires and flip them
- the flag (this one is VERY hard, lets just assume you’ll never be able to do it)
- exercise bands are like $10
- look around the house for an exposed I-beam to do pullups on
-il edit this for more later
Also a tip about getting dumbbells: try and get ones where you can add or take off weight plates yourself (rather than dumbbells that are solid all the way through, and thus they’re unchangeably ‘10kg’ or whatever). You need different weights for different exercises, and hopefully you’ll gradually need heavier weights for each exercise, so it’s not helpful to be stuck with dumbbells you can’t adjust.