Don't worry upgrading the bike, it's not that big a deal. For where you are now, durability and dependability is more important than outright performance and speed in the equipment.
With just 4-weeks of training, don't sweat it, it takes months upon months, year after year of training to get fast. You're doing just fine.
Some of those guys you see out there who are flying down the road really fast making it look like a stroll in the park have been training for years, often putting in 500-miles PER WEEK and over 10,000 miles per year.
What you want to do is educate yourself on training, Friel's book is pretty good and takes you from a novice knowing nothing, all the way up to a training regimen for beginning racers.
You'll want to mix up your training and gradually increase your mileage. One day a week, ride 15-miles. Do this for 4-5 weeks, then up it to 20-miles. Sundays are usually good for this and ride as easy and as slow as necessary to complete the distance.
After a month of this, add some intensity. On another day of the week, do some hill intervals. Find a hilly area and just ride random loops on hills of half to 1-mile in distance. Rest on the way down, recover fully and ride up another one. Just 10-miles with 5-7 hills will do wonders for your fitness.
On yet another day of the week, do some sprints. After 3-4 miles of warm-up, pick a street-sign, or a tree or some other landmark and go as fast as you can to blow up right when you hit that spot. Rest, spin easy and recover and pick some other landmark to sprint to. Do 3-4 sprints, 8-miles or so, then go home.
During all of these rides, be mindful of your pedaling motion. If you don't have clipless pedals, definitely get some shoes and pedals. Otherwise, you can scrounge up some toe-clips and straps at local shops for pretty much free. This will help tremendously over bare platforms and will give you 80% of the benefits of clipless. Then pay attention to your feet's motion and try to go in circles with even-pressure all around. You can typically double your power-output without requiring any more additional muscle-force or effort by just adding force to the dead spots of the pedal-stroke.
Watch your gearing, 70" is way too big for the speeds you're going, try 50-60inch gears instead. Don't worry about outright speed, just keep the cadence up and the speed steady and you'll be able to go farther and longer. This will improve your fitness quicker than just going fast and pushing hard.
have fun!