You need to do some practice welds. Your welds look terrible. (Not meant as an insult, just a fact.)
Make up some test pieces, then cut them in half lengthwise. (A band saw would be good for that.) then put one end in a vice, and get out your BFH. Try to break it. That is the only way to see how much penetration you are getting. Here is a test weld that I did before doing some frame welding. The first piece I made broke easily. I didn't have enough bevel on the ends of the tube and got poor penetration. This piece I could not break or bend. You can see the marks from gripping it in a vice, and beating it with a hammer. It barely distorted it.
You didn't say what kind of welder you are using. I find stick welding to be the most difficult. I also hate stick welding because it make a lot of smoke and makes a HUGE mess all over the shop.
I mostly do TIG welding with an inexpensive DC TIG welder. It is slower, but more controllable and I am able to get much higher quality welds. It's a little more expensive to get started, but I paid less than $300 for my DC TIG. You also need an argon tank, flow regulator, and a box of thoriated tungsten electrodes.
Second to TIG would be oxy/acetylene welding. It exposes your frame to a lot more heat, but it is more easily controlled.
There is also MIG. I have tried MIG and don't like it much. It makes less attractive welds.
So, enough about your welding. Time to discuss proper gussets. You have introduced bending stresses in sections of tube that previously were designed for primarily compression and tension stresses. At the very least, you will get frame flex that will impair handling. You may eventually get fatigue cracks and total, catastrophic failure. NOT GOOD!
Take your cue from the factory gussets, and make sure your new gussets transfer the stresses back to where they were before. You don't need heavy gusseting like 1/4" steel, but your gussets new to be long enough to resist the bending forces you have introduced and transfer those loads back to where the original frame member that you removed transmitted them. In fact, 1/4" gussets would be a bad idea, because it's so much heavier than the walls of the tube, that you would have a hard time getting enough penetration into the gusset without burn through on the tube.
I would suggest that you cut out all of your work so far, and make new pieces. Tack weld it together, then take it to a welder and have it done by a pro. You are a long ways away from being ready to weld your own frame.