Converting a VHS to DVD is easy and anyone with a VHS player and a DVD recorder can do it.
Simply hook the two units together hit play on the VHS and record on the DVD and record the VHS.
A compo VHS/DVD recorder unit with dub does this with the push of one button.
Now to the question of lines in the VHS tape there are two possibilities:
1. The lines are caused by your VCR not tracking properly, in that case your tape played in any other VCR will show up without the lines . Thus the above system will work and your DVD will not have the line, it will have any in perfections that were on the VHS tape.
FYI the tape is 14 years old NOT excessive for a VHS tape but it is certainly not new.
The VHS conversion in this case can be done by most photo processing stores that offer this service, including Wal-mart and Walgreen's. You will get a copy of the VHS tape with all the in perfections that are in it. They will have limited equipment to get rid of them.
2. The lines are in the tape thus it will tape will require special processing equipment to get rid of them. This could be as simple as using a good time base corrector or it may involved special digital frame by frame processing.
If the problem is with the tape only a professional camera store or video recording studio would have the necessary equipment to get rid of them.
You need to try the tape in another VHS player and get an idea if it is the tape or not.
Option 1 is cheap and easy. Option 2 can get expensive, and only they could quote a price of what it would cost after they see the image on the tape and the exact nature of the problem