First, when you bring your DWG into ArcMap, there is no need to convert it to another file type, unless you are trying to edit the linework. You can view the file and change the way it looks without doing any conversion.
(If you do need to edit it, right-click on the Polyline layer in the DWG and choose "Data" and "Export Data." You can then export it to a shapefile, so you will be able to edit the linework.)
Once you bring your DWG into ArcMap, you can change the symbology (ie. the way it looks and which layers are displayed.) The default symbology is by CAD unique entity values (Linetype, Color, LineWt.) If you click the "+" next to the Polyline you can see this. Here's an example from a DWG I'm using:
To change how it is displayed, right click on the Polyline layer and go into the Properties. Then choose the Symbology tab. Choose "Unique Values" under "Categories" and then choose "Layer" from the drop down box. At the bottom, you can choose "Add Values..." and pick the layers you want, or choose "Add All Values" and then remove the ones you don't want. Here's what the dialog box looks like:
You can change the color of individual layers or make them "no color" so that they don't show. This still doesn't take away all of the problems of using DWGs in ArcMap, but I've found it a good enough work around for much of what I use them for.
If you find this doesn't meet your needs, I would recommend converting to a shapefile like I mention above, so you can do more editing of the data (including cleaning out any text box leaders, etc. that appear as polylines).