Taking this by the numbers or devices might help. There are 4 things going on here from what I am reading, you want a new digital HDTV, you want to watch stuff from your VCR on the new TV, you want to still record shows on your VCR, and finally you want to play your existing DVD's on your new TV.
Let's go with the easy one first, the DVD player, that will hook to your new TV and no converters are needed, just make sure your TV has inputs that match the outputs on your VCR. (The DVD player should have at minimum a composite video and L/R audio outputs, if you put your brand and model number up I can likely tell you what input you will need on your new TV.)
Next is the VCR, to record broadcast digital on your analog VCR you will need a converter box to convert digital to analog, they are cheap enough so should not be an issue, using it effectively is another matter. You will have to set the channel to record on the converter before recording and then set you VCR for whatever time you want it to start, the downside is there is no way to change channels automatically from your VCR so unless you are around to change the channel between recordings you will be able to record from one channel only while you are not there.
The TV really has no part in the ability of the VCR to record shows, that is between the VCR, digital converter and the antenna.
The last part, watching your VCR on your new TV that is the same as the DVD player, just have to make sure the TV has the right kind of inputs and that will be Composite Video and Left/Right Audio, so to have both the VCR and DVD player you will want two composite inputs on your TV.
Now for the TV, you will not need any coverters for your TV, the over the air digital broadcasts will tune in just fine native, however before you dump your cable you might want to find out just what channels you can really get over the air, for example in my area I get 6 channels I can pull in from an antenna, not a lot of selection.
The only real trick with the TV is just getting one that has the inputs you need for what you plan to hook to it, all of them will have HDMI, which you do not need yet, and most will have at least one analog input, you want 2.