adammo93, you should definitely upgrade your entire system; as your current system is towards the low-end spectrum in terms of specifications.
I would ignore what everyone else has stated above; yes an i3 or i5 is obviously ideal, however they are not within your budget (no offence). Regardless that is not a problem, you don't need an i3 or above, you can always upgrade to those processors later when you save more funds; even the Haswell Celeron will provide a major, I repeat major performance boost over your current system. The integrated graphics of the Haswell Celeron will also completely annihilate your current graphics.
If you're willing to replace your current motherboard, processor, RAM and PSU and use your current case; you can build something within your budget which will provide a significant boost in performance over your current system. For the record, your current motherboard is on socket 775 and will accept Core 2 Duo processors, regardless I wouldn't spend any more money on your current motherboard. Your motherboard also accepts SATA hard drives and optical disc drives; I'm assuming you're using IDE however? If I'm incorrect and you are NOT using the IDE connectors, please reply back and inform me (if this is the case, you must be using SATA). Once you reply with your answer, I will update the build below to accommodate SATA drives if you're currently using IDE.
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Celeron G1820 2.7GHz Dual-Core Processor ($47.99 @ Mwave)
Motherboard: ASRock H81M-DGS R2.0 Micro ATX LGA1150 Motherboard ($46.99 @ Newegg)
Memory: G.Skill AEGIS 4GB (1 x 4GB) DDR3-1600 Memory ($39.99 @ Newegg)
Power Supply: Corsair Builder 430W 80+ Bronze Certified ATX Power Supply ($39.99 @ Amazon)
Total: $174.96
Prices include shipping, taxes, and discounts when available
The above build includes a new processor, motherboard, RAM, and a PSU (power supply). I would use your existing computer case, case fans and drives (if they're SATA); if you had a larger budget, I could have accommodated more components and even a new case.
The processor I've selected above is at least approximately 70% times better in performance than your current CPU, and the integrated graphics (Intel HD) will provide you better performance than your existing card. I have also included 4GB DDR3 RAM (your current system uses the elder DDR2), and a better PSU. The PSU should allow you to upgrade your system in the future, if you were to save up more funds and purchase a graphics card.
P.S. If you were to upgrade the build I made for you in the future (if you were to save up more funds), you could even upgrade to an i3, i5, i7 or Xeon processor. The motherboard will also accept a maximum of 16GB DDR3 RAM, in addition to supporting SATA3 and USB 3.0.
All the best.