Hi Kaz !
The response time is based only on the distance between you and your server. You can reduce DNS response time or page loading time for a web server, but there is no way for me to reduce a response time for a ping.
I would take a fast VPS or dedicated server with more CPUs to help your site load faster, of course is you need to use CDN like suggestions in this thread. Server response time is really important for my sites, If you can pass this point in PageSpeed Insights from Google, your website has got an advantage from search engine optimization.
DNS response time is one among several other factors that affects your website speed. If you're concerned about the website speed, I would say you'll need to first look at optimizing the server on resources and based on your application.
I've worked with a lot guys who had great configuration servers, however software limits are set low and then complain about the speed. I'd say its optimization comes in top priority.
Did you mean for ping test or global dns response time found on page speed tests?
For ping is explained by VirtuBox for page speed tests that mean that your website get some data from other servers and that is how your DNS response time get bigger.
I think use of tool like gtmetrix.com will give exact idea and suggest the areas of improvements. If there is anything burdensome website performance down than this tool will suggest. Second most important part is paying more to your hosting provider for hosting and hardware. This option could be expensive but will provide relief more quickly and conveniently.
Have you considered that your previous VPS was hosted on a server node with routing issues? Unless your provider switched phyisical locations of the server node there is no resonable explentation unless that of a malfunction.
response time is depends upon the distance between the destination and source and also depends the number of routers between the destination and source. If you are looking for small response time, you may choose a server near to your location. routing issues also increase the response time.
Response time depends on routing, location while website loading speed is also about the server setup, the website itself, how optimized the site is and so on. If you lower the response speed, the website might load faster but not necessarily as there are other factors like the ones mentioned above.
The VPS speed depends on how many websites are you running on the VPS, the resources, RAM allocated to it. If there is adequate amount of resources available on your VPS then definitely it will run faster.
You should monitor the post-get requests coming to your server and MySQL processes used to run on the server. You should use the monitoring tools for the analysis of the server traffic, requests. You should monitor the server disk space, exim queues too for the better performance. There should be plenty of space on the server drive.
Also you should optimize websites before uploading them on your server so server speed will be properly maintained.